Norwich  llniversitv  Librai^, 

NortlAfleld,  Vermont. 
Class  NO.    JT/O      Boot^No.3fJ5^ 


University  of  California  •  Berkeley 


The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/concisemathematiOOrobirich 


CONCISE 
MATHEMATICAL    OPERATIONS; 


BEING    A 


SEQUEL 


TO    THE   AUTHOR'S    CLASS   BOOKS 


WITH     MUCH     ADDITIONAL     MATTER. 


A.  WORK  ESSENTIALLY    PRACTICAL,   DESIGNED  TO   GIVE   THE   LEARNER   A   PROPER   APPRE* 

CIATION  OF   THE    UTILITY   OF    MATHEMATICS  ;    EMBRACING   THE    GEMS   OF 

SCIENCE    FROM    COMMON   ARITHMETIC,   THROUGH    ALGEBRA, 

GEOMETRY,   THE    CALCULUS,   AND   ASTRONOMY. 


BY  H.  N.  ROBINSON,  A.  M. 

FORMERLY    PROFESSOR  OF   MATHEMATICS  IN  THE    UNITED   STATES   NAVY  J  ATTTHOR   OF 

ARITHMETIC,   ALGEBRA,   NATURAL   PHILOSOPHY,    GEOMETRY, 

SURVSYING,   ASTRONOMY,   ETC.  ETC.    ETC, 


CINCINNATI: 

JACOB  ERNST,  112  MAIN"  STREET. 
1854. 


Entered  according  to  Act  of  Congress  in  the  year  1854,  bj 

H.  N.    ROBINSON, 

In  the  Clerk's  Ofl&ce  of  the  District  Court  of  the  United  States 

for  the  Northern  District  of  New  York. 


PREFACE. 

This  book  is  not  designed  to  teach  Mathematical  Principles,  but  to  apply 
and  enforce  them.  It  contains  collections  and  groups  of  mathematical  prob- 
lems which  show  the  utility  of  science,  and  place  its  fruits  in  the  foreground. 

Let  no  one  expect  to  find  any  close  connection  between  the  different  parts 
of  this  book,  or  even  in  any  one  part  of  it.  System  and  connection  is  essen- 
tial in  every  theoretical  work,  but  it  would  be  as.  absurd  to  look  for  it  here 
as  to  look  for  a  composition  in  a  dictionary. 

That  there  is  need  for  such  a  work  as  this,  all  would  be  convinced  who 
could  see  but  a  tenth  part  of  the  letters  that  every  accommodating  mathema- 
tician is  constantly  receiving,  requesting  the  solution  of  problems  or  the 
exposition  of  principles. 

Indeed,  much  important  matter  to  be  found  in  this  volume,  has  been  sug- 
gested and  brought  to  the  immediate  notice  of  the  author  by  letters  received 
requiring  his  aid  ;  and  to  save  the  trouble  of  answering  such  letters  in  future 
was  one  inducement  to  publish  this  work. 

There  is  a  great  deal  of  perfectly  barren  mathematical  knowledge  in  this 
country;  particularly  among  those  who  have  studied,  not  for  science,  but  for 
a  diploma. 

Not  unfrequently  do  we  meet  persons  who  can  demonstrate  many,  if  not 
all  the  elementary  problems  in  common  Geometry,  who,  at  the  same  time, 
cannot  make  the  least  application  of  them,  and  who  seem  to  be  unaware  that 
they  were  ever  intended  for  any  practical  use. 

Knowledge,  so  confined  and  abstract,  is  of  doubtful  utility,  even  as  a 
mental  discipline.  Unless  we  take  in  a  broad  expanse,  and  unite  both  theory 
and  practice,  we  perceive  nothing  of  the  beauties  of  the  Mathematics.  De- 
tached propositions  and  abstract  mathematical  principles,  give  us  no  better 
idea  of  true  and  living  science,  than  detached  words  and  abstract  grammar 
would  give  us  of  poetry  and  rhetoric.  Small  acquirements  in  the  Mathe- 
matics serve  only  to  make  us  timid,  cautious,  and  distrustful  of  our  own 
powers — ^but  a  step  or  two  further  gives  us  life,  confidence,  and  power. 

The  efforts  of  the  great  mass,  who  attempt  the  study  of  the  Mathematics, 
are  very  inefficient  and  feeble,  because  the  motive  is  not  sufficiently  pointed 
and  pressing.     They  study  for  the  discipline  of  mind. 

]N'ow,  we  venture  to  assert,  that  those  who  study  for  any  object  so  indirect 
and  indefinite,  can  never  be  decidedly  successful.  And  those  who  teach 
with  no  other  view  than  giving  discipline  to  the  minds  of  their  pupils,  never 
more  than  half  teach.  The  object,  and  the  only  object,  should  be  to  under- 
stand the  subject  studied,  and  if  that  understanding  is  attaiiied,  the  highest 
mental  discipline  that  the  subject  can  yield,  will  surely  come  with  it. 

iii 


iv  PREFACE. 

Let  a  person  undertake  the  study  of  any  science,  Trigonometry  for  exam 
pie,  with  no  other  object  than  the  discipline  of  the  mind,  and  our  word  for  it, 
the  science  will  come  to  him  with  the  utmost  diflficulty  ;  and  however  long 
he  may  study,  the  spirit  of  the  science  will  never  find  a  lodgment  with  him. 
But  let  him  be  determined  to  understand  it,  for  the  purpose  of  being  an 
architect,  an  engineer,  or  a  navigator,  and  all  is  changed — beauties  are  now 
seen  where  none  were  discovered  before,  and  the  student  is  now  sensible  of 
possessing  both  knowledge  and  mental  discipline. 

Let  a  person  commence  Astronomy,  simply  with  a  view  to  mental  disci- 
pline, and  when  will  he  obtain  a  sound  knowledge  of  that  science  ?  We 
answer,  never.  But  let  him  commence  the  study  with  a  determination  to 
understand  it,  and  his  efforts  will  be  well  directed,  and  science  will  come  to 
him  with  ease,  and  with  it  will  come  a  discipline  of  mind,  the  most  pure 
and  lasting  that  man  can  attain. 

There  is  another  erroneous  impression  which  serves,  as  far  it  goes,  to  ob- 
struct the  progress  of  sound  mathematical  learning  in  this  country.  It  is  a 
vague,  yet  general  idea,  that  Arithmetic,  Algebra,  Geometry,  Trigonometry, 
and  the  Calculus,  are  distinct  and  separate  sciences,  and  each  is  to  be  learned 
by  itself  and  then  carefully  laid  aside.  The  truth  is,  they  are  but  diJOferent 
sections  of  the  same  science,  and  each  one  in  turn  may  be  used  to  illustrate 
the  other ;  and  studied  as  a  whole,  under  the  direction  of  a  philosophic 
t^^acher,  the  labor  of  acquisition  would  be  very  much  reduced. 

Were  we  to  say  nothing  in  respect  to  our  method  of  treating  the  square 
and  cube  roots  in  this  volume,  the  mere  arithmetician  would  undoubtedly 
depreciate  it.  He  will  perhaps  still  regard  the  method  as  unscientific, 
and  call  it  a  mere  "  cut  and  try"  operation  ;  but  when  he  finds  the  same  thing 
in  Geometry,  and  there  finds  lines  which  may  represent  all  the  different 
factors  in  any  case,  and  sees  the  geometrical  reason  why  the  exact  square  root 
is  always  a  little  less  than  the  half  sum  of  two  unequal  factors,  he  must  then 
admit  that  the  cut  and  try  method  is  not  very  unscientific  after  all.  The 
truth  is,  in  the  hands  of  those  who  can  take  the  geometrical  view  of  it,  and 
who  can  use  it  with  judgment,  this  method  is  as  scientific  as  any,  and  in 
many  cases  far  more  practical  than  the  common  rilles. 

The  first  principles  of  Geometry  are,  to  a  certain  degree,  abstract ;  but  the 
application  of  Geometry,  as  appears  in  this  work,  is  far  from  being  so  ;  and 
he  must  be  a  very  practical  mathematician  who  cannot  find  something  here 
to  amuse,  to  interest,  or  to  instruct  him. 

To  the  subject  of  finding  sines  and  cosines,  both'natural  and  logarithmic, 
for  every  minute  of  the  quadrant,  we  call  special  attention — as  strict  attention 
to  that  subject  in  all  its  bearings,  will  so  readily  impress  upon  the  mind  of 
a  learner,  the  importance  of  theoretical  Geometry. 

To  the  practial  application  of  Interpolation,  we'also  call  attention.  Some 
problems  in  Mensuration  and  Plane  Trigonometry  will  be  found  very  inter- 
esting to  those  who  possess  a  taste  for  the  Mathematics,  and  we  have  ex- 
tracted several  different  solutions  from  the  works  of  others,  to  show  how 


PREFACE.  V 

diflferently  diiFerent  persons  present  the  same  thing.  There  are  few  mathe- 
matical students  who  could  not  be  greatly  benefitted  bj  a  close  perusal  of 
Spherical  Trigonometry  and  Astronomy  as  presented  in  this  work.  Any 
person  who  has  the  outlines  of  Astronomy  and  Elementary  Mathematics  can 
here  have  a  view  of  all  the  details  of  a  solar  eclipse,  in  a  comprehensible 
shape. 

There  has  been  a  great  deal  of  unnecessary  controversy  about  the  Differen- 
tial and  Integral  Calculus,  which  we  think  can  and  ought  to  be  wiped  away. 
And  we  have  here  given  a  little  foretaste  of  what  we  shall  attempt  if  cir- 
cumstances prompt  us  to  write  a  work  on  that  subject. 

It  is  not  for  us  to  assume  that  we  can  make  science  clearer  than  others, 
but  we  have  yet  to  see  the  works  of  an  author  who  has  made  the  least  attempt 
to  show  the  simple  elementary  nature  of  this  science.  They  at  once  commence 
with  the  definition  of  constants  and  variables,  and  then  direct  what  to  do. 

We  have  yet  to  see  the  first  book  that  expends  a  word  in  giving  an  idea  of 
what  the  Calculus  is,  or  what  is  the  utility  and  object  of  the  science,  and  we 
charge  more  than  half  the  obscurity  to  this  fact  alone  :  hence  we  could  not 
forbear  being  a  little  elementary  when  we  came  to  that  subject,  and  we  leave 
it  to  those  readers,  who  have  fonnerly  studied  other  works  on  this  science, 
to  say  whether  we  have  or  can  dispel  any  of  the  obscurity  that  has  so  long 
hovered  around  it. 

All  sciences  are  obscure  until  they  are  applied.  Even  Arithmetic  would 
be  so  in  the  abstract,  and  being  alive  to  this  fact  we  have  extended  the  ap- 
plication of  the  Calculus  to  more  subjects  than  we  have  hitherto  observed  in 
other  works.  For  example,  see  the  method  of  clearing  lunar  distances,  and 
the  use  we  made  of  the  same  principle  in  computing  an  eclipse. 

But  neither  in  the  Differential  nor  the  Integral  Calculus  do  we  pretend  to 
be  any  thing  like  full  or  perfect,  even  for  a  work  of  this  kind. 

We  have  only  thrown  out  a  few  practical  remarks  and  problems,  in  our 
own  unique  manner,  more  to  learn  what  is  desired,  and  what  can  be  appre- 
ciated, than  for  any  thing  else. 

"When  we  commenced,  we  did  not  intend  to  produce  so  large  a  volume;  it 
grew  on  our  hands;  but  we  believe  that  this  result  will  not  be  regretted 
by  generous  patrons. 


CONTENTS. 

PART  I.— ARITHMETIC. 

SECTION   I. 

Introduction,  13 14 

The  Philosophy  of  Multiplication  and  Division, 14 18 

Canceling, 19 ^ 

Proportion, 23 ^29 

Cause  and  Eflfect, 26 ^29 

SECTION   II. 

Exchange, -29 

Compound  Fellowship, 30 32 

Problems  in  Mensuration  and  the  Roots, 32 34 

SECTION   III. 

Powers  and  Roots, 35        16 

Alligation  Alternate, 46 48 

Position, 48— —50 


PART  II.— ALGEBRA. 

SECTION   I. 

Simple  Equations, 51 59 

Problems  Producing  Simple  Equations, 59 67 

Interpretation  "of  Negative  Values, 67 68 

Finding  and  Correcting  Errors, 69 71 

Pure  Equations, 72 80 

Questions  Producing  Pure  Equations, 80 84 

vii 


viii  CONTENTS. 

SECTION   II. 

Quadratic  Equations, 84 89 

Special  Equations  in  Quadratics, 89 ^96 

SECTION   III.  I 

Quadratic  Equations  containing  more  than  one  Unknown 

Quantity, 97 108 

UnTfTought  Examples, 109 110 

SECTION  IV. 

Problems  producing  Quadratic  Equations  containing  more 

than  one  Unknown  Quantity, 110 116 

Problems  Selected  from  Various  Sources, 116 123 

SECTION   V. 

Problems  in  Proportion  and  Progression, 123 124 

Geometrical  Progression  and  Harmonical  Proportion, 125 129 

Proportion, 129 130 

Additional  Problems, 130 138 

SECTION  VI. 

Solution  of  Equations  of  the  Higher  Degrees, 138 161 

H'ewton's  Method  of  Approximatic«i, 138 140 

Horner's  Method, ^ 140 157 

l^ew  and  Concise  Formula  to  find  Approximate  Roots  in 

Quadratics, 143 144 

Cubic  Equations, 148 152 

The  Combination  of  Roots  in  the  Fca-mationof  Coefl5cients,.158 161 

Recurring  Equations, 161 166 

SECTION  VII. 

Indeterminate  Analysis, 166 181 

Properties  of  Numbers, 166 170 

Indeterminate  Problems, , . . , 171 181 


CONTENTS.  k 

SECTION   VIII. 

To  determine  the  Number  of  Solutions  that  an  E<iuation  in 

the  form  AX-^BY=C,  will  admit  of, 181 185 

SECTION   IX. 

Diophantine  Analysis, 185        192 

SECTION   X. 

Double  and  Triple  Equalities, 192 200 

Double  Equalities, 193 195 

Triple  Equalities, 195 200 

Application  of  the  Diophantine  Analysis, 201 ^202 


PART  III.— GEOMETRY. 

SECTION    I. 

Geometrical  Theorems, 20.3 ^218 

Geometrical  Constructions, 219 ^223 

Constructions  for  finding  the  Square  Roots  of  Numbers,. .  .221 ^223 

Geometrical  Problems  requiring  the  aid  of  Algebra, 224 ^241 

Numerical  Problems, 240 ^241 

SECTION  II. 

Trigonometry, • 243 ^268 

The  Most  Concise  and  Practical  Method  of  finding  the 

Circumference  of  a  Circle, 242 246 

Interpolation  :  Its  Utility  in  finding  the  Sines  and  Cosines 

of  each  Minute  of  the  Quadrant, 246 248 

On  finding  Logarithmic  Sines  and  Cosines, 248 251 

Solution  of  Trigonometrical  Problems, 251 255 

Problems  in  Mensuration, 256 262 

Theorems  on  pages  219  and  220  of  Robinson's  Geometry,  262 267 

Additional  Theorems, 267 ^268 


CONTENTS. 

SECTION   III. 


Problems  in  Spherical  Trigonometry  and  Astronomy, 269 ^272 

Problems  from  Page  215,  Robinson's  Geometry, 273 ^285 

Other  Problems  of  like  kind, 285 ^286 


PART  IV.— PHYSICAL  ASTRONOMY. 

kbpler's  laws, 

Kepler's  Laws, 287 ^289 

Propositions  from  Robinson's  Astronomy,  page  146, 289 291 

Increase  of  the  Moon's  Periodic  Revolution, 292 293 

To  find  the  Position  of  a  Planet  as  seen  from  the  Earth,  293 297 

SOLAR   ECLIPSES. 

The  Computation  of  the  Eclipse  of  May  26th,  1854,  for 

the  Lat.  of  Burlington,  Vt.,.\ 297 313 

The  Elements, 297 398 

Tables  for  Correcting  the^  Elements, 298 299 

To  determine  the  time  of  the  beginning  of  the  Eclipse, 

and  the  Place  at  which  it  will  be  Central, 299 301 

Parallax  in  Altitude, 301 

Distance  between  the  Centers, 302 305 

Point  of  first  Contact, 305 

Greatest  Obscuration, 305 309 

The  End  of  the  Eclipse, 309 313 

Summary, 313 

Correspondence, '. . .  .313 314 


CONTENTS.  xi 

THE  CALCULUS. 

DIFFERENTIAL    CALCULUS. 

Differential  Calculus, .' 315 340 

Subject  Defined, 315 

Logarithmic  Differentials, 317 321 

Circulai-  Functions, 322-^ — 330 

Differential  Expressions  for  Trigonometrical  Lines, 322 

Examples  showing  the  Utility  of  the  Calculus, 322 326 

Additional  Examples  in  Circular  Functions, 326 330 

Lunar  Observations, 330 333 

Maxima  and  Minima, 334 34fr 

INTEGRAL    CALCULUS. 

Introductory  Remarks  and  Exercises, 340 342 

To  Integrate  two  or  more  Expressions, 342 345 

Application  of  the  Integral  Calculus, 345 350 

To  find  the  Value  of  a  Semicircle  to  Radius  Unity, 345 346 

Application  of  the  Calculus  to  Surfaces, 346 348 

To  Solids, 348 350 

Two  Integrations  from  Poisson's  Mecanique, 350 353 


MATHEMATICAL  TABLES. 

TABLE    I. 

Logarithms  of  Numbers  from  1  to  10000, 1 20 

TABLE   II. 

Logarithmic  Sines  and.  Tangents,  also  Natural   Sines,  for 

<'very  Degree  and  Minute  of  the  Quadrant, .  .21 65 


CONTENTS. 

TABLE    III. 

Logarithms  of  Numbers  from  1  to  1 10,  including  twelve  places 

of  Decimals, 66 67 

TABLE    IV. 

Logarithms  of  the  Prime  N"umbers  from  110  to  1129,  including 

twelve  places  of  Decimals, 67 69 

Formula  for  Computing  the  Logarithms  of  N'umbers  beyond 

the  limits  of  the  Table, 69 

Auxiliary  Logarithms, 70 

TABLE    V. 

Dip  of  the  Sea  Horizon, 71 

TABLE    VI. 

Dip  of  the  Sea  Horizon  at  different  Distances  from  it, ^71 

TABLE    VII. 

Mean  Refraction  of  Celestial  Objects, 71 


30536 


ROBINSON'S  SEQUEL. 


PART   FIRST. 

ARITHMETIC. 

SECTION   I. 

We  shah  i)e  very  brief  in  this  work  on  the  subject  of  arithme- 
tic, only  truching  on  such  points  as  are  generally  neglected  in 
the  class  r>om. 

Formerly  all  kinds  of  problems  and  puzzles  were  to  be  found 
in  arithmetics ;  but  pure  science,  good  taste,  and  the  rapid  ad- 
vancemert  of  the  pupils,  require  that  the  works  on  arithmetic 
should  bf  concise  and  clear,  and  take  no  undue  proportion  of  the 
student's  ^ime  and  attention. 

Severo  problems  do  not  teach  science — but  science  will  subdue 
all  severe  problems,  and  we  would  use  problems  only  as  a  means 
of  elucidating  science.  Algebraic  problems,  and  problems  in 
geometry  and  mensuration,  should  never  appear  in  arithmetic, 
but  old  custom  will  not  yet  tolerate  their  expulsion. 

We  shall  pay  particular  attention  to  the  metaphysique  of  the 
science. 

Numbers  only  of  the  same  kind  can  be  added  together  or  subtracted 
from  each  other. 

Numbers  are  either  abstract  or  concrete.  Abstract  numbers 
are  unapplied  and  are  mere  numerals.  Concrete  numbers  bring 
to  the  mind  the  particular  number  of  things  to  which  they  refer. 

Arithmetic  proper,  comprises  the  system  of  notation,  and  the 
operations  to  be  performed  with  abstract  numbers  only — vnihout 
any  reference  to  their  application  whatever. 

The  application  of  arithmetic  includes  all  kinds  of  numerical 

13 


U  ROBINSON'S  SEQUEL. 

computations,  and  they  are  therefore  endless  in  variety  and  char- 
acter. 

In  the  application  of  arithmetic,  there  are  two  distinct  opera- 
tions, the  logical  one  and  the  mechanical  one;  the  thinking  and  the 
doing. 

The  undisciplined  direct  their  attention  more  to  the  doing  than 
to  the  thinking,  when  it  should  be  the  reverse;  and  nearly  all  the 
efforts  of  a  good  teacher  are  directed  to  make  his  pupils  reason 
correctly. 

If  a  person  fails  in  an  arithmetical  problem,  the  failure  is  always 
in  the  logic,  for  false  logic  directs  to  false  operations,  and  true  logic 
points  out  true  operations. 

Abstract  arithmetic  we  shall  not  touch,  except  when  necessary 
to  illustrate  a  point  before  us. 

With  these  introductory  remarks  we  commence  with  the  follow- 
ing principles : 

1.  Multiplication  is  the  i^epetition  of  one  number  as  many  times 
as  there  are  units  in  another. 

This  is  general,  whether  the  numbers  be  large  or  small,  whole 
or  fractional.  The  mles  in  whole  numbers  and  in  fractions,  apply 
to  the  mechanical  operations  only,  and  not  to  the  one  fundamental 
principle. 

2.  When  the  multiplicand  and  multiplier  are  both  abstract  num- 
bers, the  product  is  abstract,  or  a  mere  numeral  without  a  name. 

3.  iVb  two  things  can  be  multiplied  together. 

A  multiplicand  may  have  a  name,  as  dollars,  yards,  men,  <fec. ; 
then  the  multiplier  must  be  a  mere  numeral,  and  the  product  will 
have  the  same  nam>e  as  the  multipHcand. 

4.  Division  is  finding  how  many  times  one  number  can  he  sub- 
tracted from  another  of  the  same  hind. 

There  are  other  definitions  to  be  found  in  books,  which  do  very 
well  in  the  main,  but  this  is  the  only  truly  logical  definition  I  can 
find.  Division  should  never  be  considered  in  the  light  of  sepa- 
rating a  number  into  parts,  for  this  is  not  true  in  all  cases,  and 
confusion  often  arises  in  fractions  by  this  view  of  the  subject. 


ARITHMETIC.  16 

5.  Division  corresponds  to  multiplication  conversely,  when  we  take 
the  product  for  a  dividend,  the  multiplicand  for  a  divisor,  and  the 
quotient  for  a  multiplier, 

6.  In  multiplication  it  is  indifferent  which  of  the  two  factors  is 
called  the  multiplicand,  the  other  must  be  an  abstract  multiplier.  The 
name  of  the  product  (when  known)  is  an  infallible  index  to  show 
which  of  the  two  factors  is  really  the  multiplicand. 

To  illustrate  principles  three  and  six,  we  give  the  following 

EXAMPLES. 

1 .  What  will  763  pounds  of  pork  come  to  at  8  cents  per  pound? 
At  first  view,  this  example  seems  to  conflict  with  principle  three, 

for,  says  the  pupil,  we  multiply  the  pounds  by  the  price  per  pound; 
but  it  is  not  so. 

Two  pounds  would  cost  twice  as  many  cents  as  one  pound,  and 
763  pounds  would  cost  763  times  as  many  cents  as  one  pound ; 
therefore  763  is  the  abstract  multiplier  in  the  operation,  and  8 
cents  is  the  true  multiplicand,  and  the  product  will  be  cents,  as 
required. 

In  the  act  of  multiplying,  it  is  indifferent  how  the  numbers  are 
written. 

2.  Reduce  6£  135.  ^d.  to  pence. 

Here  20  and  5,  as  abstract  numbers,  must  be  multiplied  together 
and  13  added,  making  113  shillings, — but  which  of  the  two  fac- 
tors 20  or  5,  is  the  multiplicand? 

Here  nine-tenths  of  those  who  teach  arithmetic  would  call  the 
6£  the  multiplicand  and  20  the  multiplier ;  but  this  is  not  so.  A 
multiplicand  suffers  no  change  of  name  by  being  multiplied,  and 
as  the  name  of  the  product  is  unquestionably  shillings,  20  shil- 
lings is  the  multiplicand,  and  5,  as  an  abstract  number,  is  the 
multiplier,  there  being  5  times  as  many  shillings  in  5£  as  in  1  £. 

By  the  same  logic,  to  reduce  113  shillings  to  pence,  12,  the 
number  of  pence  in  a  shiUing,  is  the  true  multiplicand,  which 
must  be  repeated  113  times,  and  then  the  product  must  of  course 
be  pence  as  required. 

Dollars  can  be  divided  by  dollars,  and  by  nothing  else.  Yards 
can  be  divided  by  yards,  and  by  nothing  else,  and  so  on,  for  any 
other  thing  tha*  might  be  mentioned. 


16  ROBINSON'S  SEQUEL. 

This  fact  has  not  been  sufficiently  attended  to ;  indeed,  it  has 
scarcely  been  recognized  by  many  teachers. 

It  is  true  we  can  divide  a  number  of  dollars,  yards,  <fec.  into 
equal  parts,  but  we  do  so  indirectly,  in  point  of  logic,  while  the 
mechanical  operation  is  direct. 

That  dollars  can  only  be  divided  by  dollars  arises  from  the  fact 

that  division  is  but  a  short  process  of  finding  how  many  times  one 

<  quantity  can  be  subtracted  from  another,  and  we  can  subtract  only 

dollars  from  dollars,  therefore  we  can  divide  dollars  only  by  dollars. 

Example. — Divide  842  equally  among  6  men. 

Now  we  cannot  divide  842  by  6  men  nor  by  6 ;  but  if  we  give 
each  man  a  dollar,  that  will  require  86,  and  ^Q  can  be  subtracted 
from  842  seven  times.  Hence  we  can  give  each  man  a  dollar 
seven  times,  or  we  can  give  him  87  at  one  time.  After  the  ope- 
ration is  performed,  we  may  call  the  7,  seven  dollars,  then  the  6 
will  be  a  mere  number,  and  thus,  indirectly,  we  may  divide  842 
by  6.  Practically,  however,  all  such  operations  are  performed 
abstractly,  as  42,  6,  and  7,  taken  as  mere  numbers,  and  then 
mere  logic  decides  upon  the  names.  For  another  example. 
Divide  11£  7s,  8c?.  into  4  equal  parts. 

Lay  out  a  £  in  four  different  places — this  will  require  4j£. 
Now  we  are  to  consider  how  many  times  4£  can  be  subtracted 
from  11  £,  which  is  2  times  and  3£  over,  which  reduced  to  shil- 
lings and  7  added,  makes  67  shillings ;  this  divided  by  4  shillings 
gives  16,  and  3  shillings  over,  which  reduced  to  pence  and  8  added, 
makes  44  pence,  which  for  the  same  reason,  divided  by  4  pence 
.  gives  11.  The  operation  stands  thus : 
'  4)11£  75.  8c?.(2 


3 
20 

4)67(16 
64 

3 
12 


4)44(11 
44 


ARITHMETIC.  17 

The  first  divisor  is  4£,  the  second  4s.,  and  the  third  4d.;  but  as 
the  divisor  and  the  quotient  may  change  names,  we  may  say  2£ 
\6s.  lid  is  the  original  sum  divided  into  four  equal  parts. 
The  following  example  will  further  illustrate  this  philosophy : 
Divide  42 1£  14s.  M.  among  3  men,  5  women,  and  7  boys,  giving 
each  woman  three  times  as  much  as  a  hoy,  and  each  man  double  the 
sum  to  a  woman.     Required  the  shares  for  each, 
I  will  commence  by  giving  one  boy  1  £, 
One  woman  3£. 
One  man     6£. 
But  there  are  7  boys ;  to  give  each  1  £,  would  require     7 j^ 
5  women  each  3£  would  require  1 5£, 
3  men  each     6£,  would  require  18£ 

40£. 

Thus  going  once  round  giving  each  boy  1  £  and  each  man  and 
woman  their  due  proportion,  would  require  40  j£.  We  are  now  to 
'Consider  how  many  such  rounds  it  would  take  to  consume  42 1£. 
In  other  words,  we  must  divide  42 1£  by  40£,  When  each  boy 
can  no  longer  take  a  pound,  we  must  in  like  manner  go  the  rounds 
with  shillings,  then  with  pence,  (fee.     The  operation  is  thus: 

40£)421£   14s.  Sd.(lO 
400 

21' 
20 


40)434(10 
400 


34 
12 


40)416(10| 
416 
Here  then  it  is  clear  that  each  boy  must  have  \£  ten  different 
times,  or  which  is  the  same  thing  in  effect,  10£  at  one  time.  Hence, 
simply  changing  the  names  of  the  divisor  and  quotient, 
Each  boy  is  to  have  10£  10s.   lOfc?. 

Each  woman  3  times  as  much,  or  31£  12s.     I^xl. 
And  each  man  63£     >5*>     2f'i, 

2  " 


18  ROBINSON'S  SEQUEL. 

We  will  give  but  one  more  example  to  show  that  the  divisor 
and  dividend  must  be  of  the  same  name. 

The  moon  describes  an  arc  in  the  heavens  of  197°  38'  45"  in  15 
days;  how  great  an  arc  will  it  describe  in  1  day? 

The  fifteenth  part  of  197°  38'  45"  is  obviously  the  sum  required, 
but  how  will  the  number  15  measure  197  degrees?  If  we  drop 
the  name  of  degrees  and  say  197,  then  we  can  divide  it  by  15, 
and  this  is  the  usual  way — during  the  operation  all  names  are  prac- 
tically destroyed — and  after  the  operation  is  over,  the  proper  name 
is  given  according  to  the  logic  or  philosophy  involved  in  the  ques- 
tion, and  it  is  in  this  logic  or  philosophy  where  the  unthinking  fail, 
if  they  fail  at  all. 

We  may  also  solive  this  problem  by  conceiving  the  moon  to 
move  15°  in  one  day,  and  then  dividing  197°  38'  45"  by  15°,  we 
shall  obtain  an  abstract  number,  each  unit  of  which  corresponds 
to  a  day.  Then  changing  the  names  between  the  divisor  and  the 
quotient,  16°  will  become  an  abstract  number,  and  the  quotient 
will  be  degrees,  ^(Scc.  as  required. 

15°)  197°  38'  45"(13 
15 


47 
46 


60 


J5)158(JO 
150 


8 

60 


16)525(36 
45 


76 

76 
Our  first  divisor  was  15°,  second  15',  third  15",  but  by  making 
these  abstract  numbers,  the  quotient  will  become  13°  10'  35",  the 
answer. 


CANCELING.  19 


€ANCEL,ING. 


Within  a  few  years  the  subject  of  canceling  has  been  brought 
to  the  special  notice  of  teachers  and  others,  and  like  every  other 
improvement,  it  has  been  opposed  by  some,  and  looked  upon  with 
distrust  and  indifference  by  others.  But  still,  it  being  a  real  and 
substantial  improvement,  it  is  working  its  way ;  and  even  at  this 
day  intelligent  pupils  are  astonished  that  teachers  should  op- 
pose it  as  they  sometimes  do.  Indeed,  that  teacher  who  would  in 
any  degree  discountenance  cancellation  should  be  dismissed  at 
once  from  the  class  room. 

Some  few  educationists  had  private  reasons  of  a  pecuniary 
nature  for  opposing  cancellation;  but  the  chief  opposition  arose 
from  the  disinclination  of  persons  to  break  into  old  habits. 

Cancellation  does  not  change  the  process  of  reasoning  on  a 
problem,  iDut  it  requires  a  more  general  perception  at  a  glance, 
and  more  rapidity  of  thought,  than  the  old  methods;  hence  the 
naturally  dull  did  and  do  yet  oppose  it  as  a  matter  of  course. 

The  architect  makes  the  design  of  a  proposed  building  on  paper, 
represents  it  inside  and  out,  estimates  the  cost,  suggests  changes 
and  improvements,  and  has  it  all  in  his  mind  before  a  stick  of  timber 
is  prepared,  or  any  serious  labor  commenced.  It  is  economy  to  do  so. 
An  arithmetician  should  do  the  same ;  he  should  be  able  to  repre- 
sent what  he  proposes  to  do,  on  paper,  look  at  it  and  consider  it 
fully  before  he  commences  real  labor.  It  is  economy  to  do  so,  for 
then  he  may  see  counter  operations  that  will  cancel  or  abridge 
each  other.  Desirable  as  all  this  is,  it  is  rarely  thought  of; — 
no  sooner  is  an  operation  decided  upon  than  the  operator  hastens 
to  perform  it,  without  thinking  further  until  that  is  done.  He  then 
decides  upon  another  step  and  performs  it, — then  another,  and  so 
on  through  the  problem. 

Now  as  a  general  rule  we  would  have  each  step  of  an  operation 
distinctly  indicated  before  it  be  performed,  and  then  examined  as 
a  whole,  the  same  as  an  engineer  would  examine  a  map,  or  an 
architect  the  plan  of  a  building.  We  give  the  following  exam- 
ples to  exercise  this  faculty : 


20  ROBINSON'S  SEQUEL. 

1.  A  merchaiU  bought  526  barrels  of  flour  at  $4.50  per  barrel^ 
and  paid  in  cloth  at  $2.25 per  yard.     Bow  many  yards  did  it  require? 

Ans.  1052. 

The  following  is  the  common  method  of  thought  and  operation. 
We  must  find  what  the  flour  will  amount  to,  and  as  soon  as  that 
thought  is  defined,  the  operator  commences  the  multiplication. 
When  that  is  done,  then  comes  the  thought  about  the  cloth,  and 
it  is  decided  to  divide  the  amount  by  $2.26  for  the  required  re- 
sult, and  the  operation  stands  thus : 

526 
460 


26300 
2104 


225)236700(1052 

225 


1170 
1126 


450 
450 

Now  we  would  not  change  the  direction  of  the  thought  in  the 
least,  but  we  would  have  it  continued  to  the  end,  and  each  opera- 
tion indicated  as  we  go  along.* 

The  map  of  the  whole  operation  stands  thus: 
526-450 
225  ~ 

Here  is  a  fraction,  the  numerator  consisting  of  two  factors,  the 
denominator  of  one  factor  which  is  contained  twice  in  460. 
Hence  twice  526  is  the  required  result,  and  the  mechanical  ope- 
ration is  just  nothing  at  all. 

♦When  two  numbers  are  to  be  multiplied  together,  we  write  them  with  a  point 
between,  thus  4.6  indicates  4  multiplied  by  6.    If  this  is  to  be  divided  by 

any  other  number,  say  .3,  we  would  write     -1-.     When  two  numbers  are  to 

be  added,  we  write  (+)  plus  between  them;  when  one  is  to  be  subtracted, 
we  write  ( — )  minuB  before  that  one. 


CANCELING.  n 

2.  How  much  will  540  yards  of  cloth  cost  at  3s.  4d.,  in  dollars 
at  6  shillings  each? 

The  map  of  the  operation  is  thus:  — —- ?-. 

This  reduces  to  90 'SI-.  Multiply  one  factor  by  3,  and  divide 
the  other  by  3,  (which  will  not  change  the  value  of  the  product), 
then  30-10=300,  the  result. 

N.  B.  In  this  work  we  do  not  pretend  to  explain  prime  and  composite  num- 
bers, what  numbers  will  cancel  with  each  other,  and  what  will  not.  These 
things  must  be  learned  elsewhere. 

3.  At  \9.\  cents  per  pound  what  must  he  paid  for  four  boxes  of 
sugar,  each  containing  136  pounds? 

Map  of  the  operation,  = =63  dollars. 

4.  What  will  one  hogshead,  or  63  gallons  of  wiTie,  cost  at  Q\ 
cerUs  a  gill?  Ans.  ^126. 

Map  of  the  operation,  =126. 

The  multiplication  indicated  in  the  pumerator  reduces  the  63 
gallons  to  gills,  and  as  6|-  cents  is  one-sixteenth  of  a  dollar,  we 
divide  by  16,  which  cancels  the  product  of  the  fours  in  the  nu- 
merator and  leaves  63  to  be  doubled  for  the  result. 

5.  At  1^'  cents  a  gill,  how  many  gallons  of  cider  can  be  bought 
for  $24?  Ans.  50. 

24* 100 
Map  of  part  of  the  operation =:the  number  of  gills, 

or    24-200      ,,       .,,  *  '        ' 
=the  gills. 

HT        r  .1        1    1  .•        24-200       200     ^^    . 

Map  of  the  whole  operation, = =50  Ans. 

^  ^  3-4-2-4       4 

6.  Jf  a  man  travel  39  miles  20  rods  in  a  dag,  how  many  days 
leill  be  required  to  traverse  25000  miles?  Ans.  640. 

As  320  rods  make  a  mile,  the  following  is  the  m? 
320-25000 
320-39-1-20 


22  ROBINSON'S   SEQUEL. 

Here  numerator  and  denominator  can  be  divided  by  20,  which 

,         ,,  ,.      ,    16-25000 

reduces  the  operation  to . 

^  16-39+1 

Because  no  further  reduction  can  be  made,  this  last  indicated 
operation  must  be  performed  in  full. 

In  all  cases,  whether  reduction  can  be  made  or  not,  we  would 
insist  on  having  the  operations  first  indicated;  and  in  practice, 
nine-tenths  of  the  operations  can  be  reduced.  There  is  now  and 
then  one  that  cannot  be  reduced.  Even  when  the  plan  of  an 
arithmetical  operation  is  laid  down,  judgment  should  be  used  in 
drawing  out  the  final  result,  as  the  following  example  will  illus- 
trate: 

Required  the  value  of  the  following  expression : 

4900\2 


/43y      /144y      /4900\ 
\80/        \  95  /        \  24  / 


This  occurs  on  page  156  of  Robinson's  University  Algebra, 
and  we  have  seen  it  literally  carried  out  as  indicated,  in  several  of 
the  best  schools  in  the  country;  no  reductions  being  made  until 
after  the  numbers  were  squared ;  thus  making  a  long  and  tedious 
process. 

The  proper  way  is  to  take  the  square  root  of  the  expression  ; 

then  we  shall  have     l?.iil.l?00^ 
80      95        24 
Reducing  does  not  change  the  value  of  the  expression ;  the 
first  obvious  reduction,  is  to  divide  the  numerator  and  denominator 

43      6      490 
by  10  and  24  ;  then  the  expression  will  stand  thus,  —  •■ . . 

A  still  further  reduction  ffives  •  —  •  — =166,  nearly. 

^  2      19      1  ^ 

Now  the  square  of  166.3,  is  the  value  of  the  required  expres- 
sion. 

We  square,  because  the  square  root  was  taken  in  the  first  step. 
We  may  do  this,  because  we  have  no  where  changed  the  value 
of  the  expession,  except  in  taking  the  root. 


PROPORTION.  23 


PROPORTIOIV. 

This  manner  of  expressing  an  operation  is  most  efficacious  and 
practical  in  proportion. 

We  shall  make  no  attempt  to  elucidate  the  principles  of  propor- 
tion, our  attention  for  the  present  being  entirely  on  numerical  op- 
erations. 

EXAMPLES. 

1.  If  9,ciut.  ^qr.  ^Ub.  of  stigar,  oo&t  ^£  Is.  80?.,  what  wUl 
Shcwt.   Iqr*  cost? 

cwt.  qr.  lb.       cwt.  qr.  £,  s.  d. 

Statement.         2      3    21   :    35     i     :   :     6   1     8 
This  example  is  taken  from  an  old  but  popular  book,  in  which 
the  solution  covers  about  two  pages.     The  sugar  is  reduced  to 
pounds,  and  the  money,  to  pence.      The  result  of  the  proportion 
is  then  obtained  in  pence,  which  being  reduced,  gives  73£. 
We  do  it  thus  :  Reduce  the  sugar  to  qrs.     Then  the  proportion  is 

11|  :   141    :   :  6£  U,  Sd. 
Multiplying  the  two  first  terms  of  this  proportion  by  4,  which 
does  not  change  the  proportion,  then  we  have 
47  :   141-4  :  :  6£U.  M. 
or  1   :       3-4  :  :  6£  U.  M. 

Therefore  12  times  the  third  term  is  the  result,     73£. 

2.  If  3cwt.  of  sugar  cost  9j2  Is.,  what  will  4cwt.  Sqr.  26lb.  cost 
nt  the  same  r<ate?  Ans.  1 5£  25.  3c?. 

We  give  the  following  solution  just  as  it  appears  in  a  very  pop- 
ular book  : 


4.7 


cvi. 

cwt. 

qr. 

lb. 

£s.d. 

3 

:   4 

3 

26 

::  9  2  0 

4 

4 

20 

'  12 

19 

182 

7 
84 

7 
133 

12 

2184 

4 

4 

336Z§. 

:  bbm 

- 

:  2184 

»We  take  tire  old  scale  of  28  pounds  to  the  quarter. 


2L4  ROBINSON.'S  SEQUEL. 

336/6.  :  558/6.   :  :  2184 
558 


17472. 
10920 
10920 

3627: 

336)1218672( 
1008 

2106 
2016 

12)3627 

20)302     3rf. 
15£  2s. 

907 
672 

2352 
2352 

If  the  question  had  called  for  the  cost  of  2  pounds  more  of 
sugar,  it  would  have  called  for  the  price  of  ^wt 
Then  the  proportion  would  have  been 

3:5::  9.1£.  £    s.    d. 

Whence  the  cost  of  bcwi.  would  be  15    3      4 


For  the  cost  of  2  pounds  we  have 
3-112  :  2  ::  182  shillings. 
or     168  :  J  :  :  182 
or       84  :  1  :  r    91  :  lyV  shillings. 


2lb.  cost  1      1 


15    2     3  An^. 


3.  If  \h\  hashdt  of  dautr  co&t  Si 56^,  how  many  bushels  can  b& 
bought  for  S95|?  •  A^is.  9y\y^. 

Statement,  156.26  :  95.75"  :  r  15.625. 

When  a  statement  is  properly  made,  drop  all  names  and  ope- 
rate as  abstract  numbers;  then  the  proper  name  can  be  given  to 
the  result  by  the  rules  of  logie-,  or  rather,  the  true  name  comes 
as  a  matter  of  course.  Those  who  operate  by  rule  and  without 
thought  and  close  observation.,  would  make  very  tedious  work  of 
this. 

rru  c.-L  .•       •    *v,  95.75(15.625) 

The  map  of  the  operation  is  thus: — ^v ^ — ^. 

(156.25) 


PROPORTION.  26 

The  factor  in  the  denominator  is  10  times  one  of  those  in  the 
numerator,  therefore  the  operation  reduces  to 

^^'^1=9.515  Ans. 
10 

4.  1/240  bushels  of  wheat  can  be  purchased  at  the  rate  of  $22^ 

far  18  bushels,  and  sold  at  the  rate  of  ^33^-  for  22 1  bushels,  what 

would  be  the  profit?  Ans.  $60. 

240 •291 
18  :  240  :  :  22\  :    cost=_ — ^1 

18 

221  :  240  :  :  33|  :•    sale=?^??i 

'  22i 

Cost  =240^=^^^=20. 15=300  dollars. 
18-2         3-2 

Sale  =^:l^lll=?:^^=3. 120=360  dollars. 
221  2 

A  complete  proportion  consists  of  four  terms;  and  in  problems, 
tho  unknown  answer  is  generally  one  of  them  ;  and  were  it  not  for 
old  prejudices,  it  would  be  conducive  to  perspicuity  to  represent 
the  unknown  term  by  a  symbol,  say  x.  Then  a  problem  stated, 
would  no  longer  consist  of  three  terms,  but  of  four. 

At  first  a  young  learner  will  not  comprehend  a  symbol  nor  an 
equation,  and  his  confusion  arises  from,  the  "very  simplicity  of  the 
thing. 

Notwithstanding  the  aversion  of  learners  to  the  use  of  symbols^ 
the  aversion  must  be  avercome  before  they  can  enter  the  first  por- 
tals of  science  ;  and  a  little  firmness  on  the  part  of  the  teacher  willi 
remove  every  difficulty  in  a  very  short  time. 

When  a  proportion  is  complete,  the  ratio  between  the  first  coupr- 
let  is  the  same  as  the  ratio  between  the  second  couplet.     Thus, 

3  :  6  :  :  8  :  16. 

Here  the  proportion  is  true,  because  6  divided  by  3  gives  the  same 
quotient  as   \Q  divided  by  S. 

Such  a  trial  will  test  any  proportion. 

Suppose  in  this  proportion  that  16  is  not  known,  and  represented 
hj  XI  then  it  becomes        3  :  6  :  :  8  :  a; . 

Whence  5  =  ^.       Or,    x  =  ^t 

3      8  a 


26  ROBINSON'S  SEQUEL. 

That  is,  when  the  three  first  terms  of  a  proportion  are  given, 
the  fourth  is  found  by  multiplying  the  second  and  third  terms 
together,  and  dividing  by  the  first. 

In  any  proportion  the  product  of  the  extremes  is  equal  to  tJie 
product  of  the  means;  and  from  this  principle  any  one  of  the 
terms  of  a  proportion  can  be  found,  provided  the  other  three  are 
given. 

A  term  may  consist  of  two  or  more  factors,  and  one  of  those 
factors  unknown  :  in  such  cases,  the  unknown  factor  may  always 
be  found  from  an  equation  formed  by  the  product  of  the  extremes 
<md  means. 

Thus     3  :  6  :  :  2a;  :   16.     Whence  6-2-a;=3-16. 

r.  3-16      . 

Or  x= =4. 

6-2 

The  foregoing  is  designed  to  prepare  the  way  for  such  problems 
as  are  usually  found  under  compound  proportion,  which  we  shall 
call 

CAUSE   AND   EFFECT. 

After  several  years  reflection,  we  have  come  to  the  conclusion 
that  the  only  clear  and  scientific  method  of  presenting  compound 
proportion  is  that  of  cause  and  effect. 

It  is  an  axiom  in  philosophy  that  equal  causes  produce  equal 
effects;  a  double  cause  a  double  effect,  &c.  In  short,  effects  are 
proportional  to  tJieir  causes. 

Now  causes  and  effects  that  admit  of  computation,  that  is,  in- 
volve the  idea  of  quantity,  may  be  represented  by  numbers,  which 
numbers  have  the  same  relation  to  each  other  as  the  things  they 
represent. 

EXAMPLES. 

l.Ifl  m£n  in  12  days  dig  a  ditch  QO  feet- long,  Zfeet  wide,  and 
^  feet  deep,  in  how  many  days  can  21  men  dig  a  ditch  80  feet  long, 
^feet  wide,  and  8  feet  deep? 

Here  7  men  in  12  days  perform  84  days  work;  the  force  or 

cause  of  removing  60* 8*6  cubic  feet  of  earth,  which  is  the  effect. 

In  how  many  days  (we  say  x  days)  can  21  men  remove  80* 3* 8 

cubic  feet  of  earth?     Hence  we  have  this  proportion  : 

Cause.        Effect.        Cause.        Effect. 

7-12  :  60'8-6  :  :  21a:  :  80'3-8. 


PROPORTION.  27 

Here  is  a  case  where  a  factor  in  one  of  the  terms  is  unknown, 
and  that  factor  is  the  answer  to  the  question. 

A  proportion  is  equally  true  when  the  same  factors  are  rejected 

from  corresponding  terms.     This  is  hut  another  form  of  canceling. 

In  this  proportion,  we  observe  the  factor  7  in  each  cause,  and  the 

factor  8  in  each  effect.     Expunging-  these,  the  proportion  becomes 

12  :  60-6  :  :  ?,x  :  80-3 

Similarly  2  :  6  :  :  x  :  Q     Whence  x=2^  days,  Ans. 

2.  If  ^  men  huild  a  wall  in  12  days,  how  long  would  it  require 
20  men  to  huild  it?  Ans.  3 3  days. 

Questions  of  this  kind  are  usually  placed  under  the  rule  of  three 
inverse;  they  do  in  fact  belong  to  compound  proportion,  or  rather, 
to  cause  and  effect ;  but  the  effect  being  the  same  in  the  supposi- 
tion, and  in  the  demand,  (that  is  the  building  of  one  wall,)  it 
may  be  omitted  and  only  three  quantities  used. 

The  following  statement  banishes  all  confusion : 
Cause.    Effect.       Cause.  Effect, 
6-12    :  1   :  :    20a;  :  1 
As  effect=effect,  therefore  cause=cause,  that  is,  20a;=6*12. 

3.  i)^  4  men  in  2\  days,  worlcing  8^  hours  a  day,  mow  6f  acres 
of  grass,  how  many  acres  (ans.  x  acres,)  will  15  men  mow  in  3| 
days  hy  working  9  hours  a  day?  Ans.  40]--  acres. 

Here  the  unit  of  cause  is  one  hour's  work  for  a  man. 
C.  E.  C.  E. 

^  4-2i-8i  :  6f  : :  15-3|-9  :  x 

Multiply  the  1st  and  3d  terms  by  4,  then 

4-2i-33  :  6f   :  :   15-15-9   :  x 
Because  2|-  is  contained  in  15  six  times  ;  and  the  1st  and  3d 


terms  contain  the  factor  3 

:  therefore 

4-11   : 

6|  :  :  6-15-3 

^  Or           2-11  : 

3A-   :  :  6- 15-3 

Or            6-11   : 

10  :  :  6-15-3 

Or                 11   : 

10  :  :   15-3  : 

X 
X 
X 

a;=VV=40j^/4fW. 
The  reader  will  observe  that  we  give  but  specimen  examples ; 
one  of  a  kind :  the  preceding  one  was  given  on  account  of  the 
fractional  factors. 


28  .     ROBINSON'S   SEQUEL. 

4.   What  is  the  interest  of  $240 /or  3^  years,  at  6  per  cent  J 
This  question  simply  demands  the  effect  of  loaning  S240  for 
3^  years,  in  case  $100  in  one  year  yields  $6. 

Cause.    Effect.       Cause.      Effect. 
100-1   :  6   :  :  240-31  :  x. 

Whence  x= ~. 

100 

This  equation  shows  the  common  rule  for  computing  interest. 
That  is  : 

Multiply  the  principal  by  the  rate  per  cent.;  that  product  by  the 
time,  and  divide  by  100. 

Now  let  us  take  this  same  example  and  reduce  the  time  to 
months,  then  the  proportion  will  stand  thus  : 
100-12  :  6  :  :  240-42  :  x. 
Cast  out  the  factor  6  from  the  first  couplet,  then 
100-2  :  1   :  :  240-42  :  x. 

Divide  the  1st  and  3d  terms  by  2,  then  we  shall  have 

940-21 
100  :  1   I  :  240-21   :  x  a;=fll_— • 

100 

This  equation  shows  a  special  rule  to  compute  interest  at  6  per 
cent,  which  is, 

Multiply  the  principal  by  half  the  number  of  months  and  divide 
by  100. 

6.  WTiat  is  the  interest  of  $1248, /or  16  days,  30  days  taken  for 
a  month,  and  12  months  in  a  year  ? 

Cause.  Effect.     Cause.   Effect. 
100.  :  6  :  :  1248  :  x 
Days    360.  16. 

Divide  the  first  couplet  by  6,     then 

1248-16 
100-60  :  1  :  :  1248-16  :  x         ^=~loO^O 

This  equation  shows  a  special  rule  to  compute  interest  for  days 
at  six  per  cent,  which  is  thus. 

Multiply  the  principal  by  the  number  of  days,  divide  by  60,  and 
that  quotient  by  100. 


PROPORTION.  S9 

6.  The  interest  on  $98,  at  8  per  cent.,  was  $25.48  :  what  was 
the  time?  Arts.  3  years  3  months. 

Cause.    Effect.      Cause.     Effect.  • 

100-1   :  8  :  :  98-a;   :  25.48 

Whence         a:=?^  =3  y.  3  m. 
8-98        ^ 

This  equation  shows  the  following  rule  to  find  the  time  in 
interest  problems  when  the  other  elements  are  given: 

Rule.  Multiply  the  interest  by  \00  and  divide  by  the  product  of 
the  principal  and  rate. 

These  general  rules  refer  only  to  forms.  It  is  not  intended  that 
they  should  be  literally  followed.  In  the  last  equation  8  and 
98,  the  principal  and  rate,  are  multiplied  inform  as  they  stand, 
and  the  fraction  can  be  canceled  down. 

The  great  detriment  to  improvement  has  been,  that  both  teacher 
and  taught,  have  clung  close  to  the  letter  of  the  rules. 


SECTION   II. 

We  shall  touch  on  but  few  points  in  this  section  ;  and  only  such 
as  will  bear  on  conciseness  of  operations. 

We  give  but  one  example  in  Exchange  and  per  centage — it  is 
the  following : 

A  merchant  bought  sugar  in  New  York  at  6  pence  a  pound,  New 
York  currency  ;  and  while  on  his  hand  the  wastage  was  estimated  at 
5  per  cent.;  and  interest  on  first  cost  at  2  per  cent.;  how  many  cents 
shall  he  ash  per  pound  to  gain  25  per  cent.  Ans.  ^-^^j. 

To  reduce  pence,  IS'ew  York  currency,  to  cents,  we  must  mul- 
tiply by  If  ;  to  increase  any  quantity  5  per  cent,  we  must  multiply 
by  1^1,  and  so  on  for  any  other  per  cent.;  hence  the  index  of  the 
operation  is  as  follows  : 

6  25  105  102  125 

T  '  24  '  100  '  100  *  Too 

This  will  cancel  to  a  considerable  extent.  This  form  is  a  gene- 
ral rtile  for  all  problems  of  the  kind.  A  loss  in  any  problem, 
of  3  per  cent,  for  example,  is  brought  in  by  the  factor  jVa ,  and  so 
on  for  any  other  estimated  loss,  expressed  as  per  centage. 


30  ROBINSON'S  SEQUEL. 


COMPOUND    FEI.I.OWSIIIP. 

Under  this  head  gains  and  losses  must  be  proportioned  by  the 
products  of  capital  and  time.     We  give  a  iew  peculiar  examples. 

1 .  Two  men  commenced  partnership  for  a  year;  one  put  in  $  1 ,0C0 
at  the  commencement,  and  four  months  afterwards  the  other  put  in 
his  capital:  at  the  close  of  the  year  they  divided  their  gains  equally. 
What  capital  did  the  second  put  in  ?  Ans.  $1,600. 

For  a  mere  arithmetical  student,  who  had  never  been  tauofht 
the  use  of  symbols,  this  would  be  a  very  puzzling  problem.  We 
are  therefore  opposed  to  taking  up  the  time  of  students  with  diffi- 
cult problems,  except  so  far  as  may  be  necessary  to  show  them 
the  necessity  and-  advantages  of  symbols  and  true  science.  This 
is  a  very  good  example  to  illustrate  the  utility  of  symbols : 

Let  X  =  the  required  capital.  It  was  in  trade  8  months",  and 
as  their  gains  were  equally  divided,  therefore  the  products  of 
capital  and  time  of  each  must  be  equal ;  that  is, 

8a;=12-1000      or,    a:=??22=1500. 

2 

2.  A,  B  and  C  had  a  capital  stock  of  $5762.  A's  money  was 
in  trade  5  months,  B's  7  months,  and  C's  9  months.  They  gained 
$780,  which  was  divided  in  the  proportion  of  4,  5  and  3.  Now  B 
received  $2087  and  absconded.  What  did  each  gain  and  put  in, 
and  did  A  and  C  gain  or  lose  by  B's  misconduct,  and  how  much? 

.,     ,  780-4     780      p      780-5      ^      780-3     780 

^'s  share= = jd  = C= = 

12  3  12  12  4 

As  gains  are  divided  in  proportion  to  capital  multiplied  by  the 
time  it  is  in  trade  ;  conversely  then,  capital  must  be  in  propor- 
tion to  the  respective  gains,  divided  by  the  respective  times. 

Their  proportional  gains  are  4,  5,  3,  which  divided  by  the  times 
5,  7,  and  9,  give  j,  f,  and  }  for  their  proportional  shares  of  the 
capital. 

But  these  numbers  being  fractional  are  inconvenient.  We  will 
multiply  each  by  5-7-3  or  106,  which  gives  the  proportional 
numbers  84,  76,  36,  the  sum  of  which  (194)  may  be  taken  as 


COMPOUND  FELLOWSHIP,  ?l 

the  number  of  shares  composing  the  capital,  §5762;  and  ^'3 
capital  is  84  such  shares,  i?'s  75,  and  C's  35.     That  is, 

^'s  capital  =!6^^.     B-s=''J^t     C's  ^^I^l^- 
^  194-1  194  194 

A  and  C  gain  by  B,  §465.57-f.. 

S.  In  a  certain  factory  were  employed,  men,  woTnen,  and  boys. 
The  boys  received  3  cents  per  hour,  the  women  4,  and  the  men  6 ; 
the  boys  worked  8  hours  a  day,  the  women  9,  and  the  men  12;  the 
boys  received  $5  as  often  as  the  women  ^10,  and  for  every  $10 
paid  to  the  women  824  were  paid  to  the  men :  how  many  men,  women 
and  boys  were  there,  the  whole  number  being  59  ? 

Ans.  24  men,  20  women  and  15  boys. 
Boys.      Women.       Men. 
Sums  per  hour,        3  cts.       4  cts.      6  cts. 

No.  of  hours,     8  9  12 

Sums  paid  to  one  of  each  class,    24  36  72 

Proportional  sums  paid  to  one  of  each,  2  3  6 

The  sums  paid  to  all  of  each  class  divided  by  the  sums  paid 
one  of  each  class  will  give  the  proportional  number  in  each  class. 

.  5  .  10  24 
That  is  ^  •  -^  '  -Q  are  the  proportional  numbers  of  persons  res- 
pectively. Multiply  by  6  to  clear  of  fractions,  for  fractional 
numbers  cannot  apply  to  persons.  Then  the  proportional  numbers 
will  be  15,  20,  24,  and  as  these  numbers  make  59  they  are  the 
numbers  in  fact. 

4.  A,  B,  arid  C,  are  employed  to  do  a  piece  of  work  for  $26.45: 
A  and  B  together  are  supposed  to  do  |  of  it,  A  and  G  ~^,  and  B 
and  C  If,  and  are  paid  proportionally  to  that  supposition  :  what  is 
each  man's  share?  Ans.  A  $11.50,  J3  $5.75,    C$9.20. 

This  problem  is  algebraic,  and  the  operation  is  algebraic 
whether  the  symbols  be  used  or  not,  and  this  is  true  of  many 
other  problems  found  in  Arithmetics. 

Here  A  works  with  JB  and  with  C,  and  we  must  discover  what 
he  is  supposed  to  do,  working  alone.     It  is  done  thus : 
A+B=  f.  (1) 

-4+C  =  A  (2) 

£+C=ii.         (3) 


S2  ROBINSON'S  SEQUEL. 

By  addition,  2(  J+^+(7)=:iA_[.i^_|_t  i=^||. 

Dividing  by  2,  ^-|-i?+(7=|f.  (4)  This  equation 
shows  that  the  anaount  each  one  was  supposed  to  do  was  over 
estimated. 

Equation  (3)  taken  from  (4)  gives  A=z^% — H=^H' 

-  (2)  from  (4)      -     B=.U-H--^- 

-  (1)  from  (4)      -      (7=H-M=^V 

For  A's  portion  of  the  money  we  have  the  following  proportion: 
f  4  :  i^  :  :  26.45  :  A's  part. 

Or,       23  :   10  :  :  26.45  :  _^^1:^=  $11.50. 

23 

6.  A  person  after  doing  *}  of  a  piece  of  work  in  30  days,  calls  in 
an  assistant,  and  together  they  complete  it  in  6  days :  in  what  tlni€ 
could  the  assistant  alone  do  the  whole  work^  Aiis.  2 If  days. 

If  the  person  could  do  |  in  30  days,  he  could  do  ^  in  10  days, 
and  in  one  day  he  could  do  jV  ^^  the  whole  work.  Therefore,  it 
would  require  50  days  for  him  to  do  the  whole  work  alone. 
Again,  §■  of  the  work  being  done  |  remained  to  be  done  ;  on  this 
the  first  person  worked  6  days  and  did  /„  ^^  it-  Then  ( |  —  /o ) 
or  ^^  remained  for  the  assistant  to  do  in  6  days ;  hence  he  must 

do  aVo  ill  ^^^  day,  or in  one  day.     Therefore,  to  do  th-e 

Sly 

whole  he  w^uld  require  21f  days,  the  answer  required. 


PROBI.CM§  IN  MENSURATION  AND  THE  ROOTS. 

Mensuration  and  the  Roots  belong  to  Geometry  and  Algebra, 
but  custom  requires  that  some  practical  problems  under  these 
principles,  should  appear  in  every  Arithmetic.  We  select  such 
examples  as  will  illustrate  numerical  brevities. 

1 .  How  many  feet  in  a  board  22  inches  wide  at  one  end,  8  i^cht^ 
wide  at  the  other,  and  14  feet  long  ?  Ans.  17^  feet. 

Index  to  the  operaticm,    — 


MENSURATION  AND  THE  ROOTS.  3S 

2.  A  man  bought  a  farm  198  rods  long,  150  rods  wide,  at  832 
per  acre  ;  what  did  the  farm  come  to?  Ans.  $5940. 

T  J      *    *!,              r            198-150-32     ,^„  „^ 
Index  to  the  operation, =198-30. 

3.  If  the  forward  wheels  of  a  coach  are  four  feet  in  diameter,  and 
the  hind  wheels  5  feet,  how  many  more  times  will  the  former  revolve 
than  the  latter  in  going  a  mile,  estimating  the  diameter  of  a  circle  to 
the  circumference,  as  7.  to  22.? 

Circumference  of  the  fore  wheels,  = ;  hind  wheels, ' 

7  7 

There  are  5280  feet  in  a  mile  ;  this,  divided  by  the  circumfer- 
ence of  each  wheel  will  give  the  number  of  revolutions  of  each 
wheel. 

The  fore  wheels  revolve times. 

22-4 

5280*7 
The  hind  wheels  revolve times. 

22-6 


-n-ff  5280-7/1 

Dmerence 


22 
22-20  22 


^'(H) 


^,    ,  .       5280-7     264-7     ,„  ^     ^.    . 
That  IS     = ._=12-7=84  Ans. 


4.  The  bin  of  a  granary  is  10  feet  long,  5  feet  wide,  and  4  feet 
high;  allowing  the  cubical  contents  of  a  dry  gallon  to  contain  26 8| 
cubic  inches,  how  many  bushels  will  it  contain?         Ans.  \Q\^^-^. 

10-12-6-12-4-12     50-12-5-12-6 


Index  to  the  operation. 


1341 


5.  A  man  wishes  to  make  a  dstern  8  feet  in  diameter  to  contain 
60  barrels,  at  32  gallons  each  and  231  cubic  inches  to  a  gallon:  what 
must  be  the  depth  of  the  cistern?  Ans.  61|-  inches. 

The  diameter  of  the  cistern  is  96  inches ; 
Its  area  is  96 -96 -(0.7854.) 

60-32-231  I 


The  index  to  the  operation  is 


96 -96 -(0.7854) 
3 


34  ROBINSON'S  SEQUEL. 

6.  What  will  it  cost  to  build  a  wall  240  feet  long,  6  feet  ?dgh, 
and  3  feet  thick,  at  $2>.^5  per  1000  bricks,  each  brick  being  9  inches 
long,  4  inches  wide,  and  2  inches  thick?  Ans.  $336.96. 

Index  240'12'6'12'3'12-(3.25) 
1000-9-4-2   . 

7.  The  bung  diameter  of  a  cask  is  38  inches,  the  head  diameters 
inside  the  staves  28  inches,  and  the  length  45  inches:  how  many 
wine  gallons  will  it  contain?  Ans.  167.89-|-. 

N.  B.  The  cask  is  conceived  to  be  two  equal /rwi-^wms  of  cones 
joined  by  their  greater  diameters.     {^See  Geometry.) 

Index  to  solution  (5^^+28^+28-38),7854-45 

3'231 
Observe  that  the  decimal  0.7854  is  divisible  by  231  :  quotient 
.0034.     Therefore  we  may  have  the  following  rule  to  find  the  num- 
ber of  wine  gallons  in  a  cask  : 

Rule.  To  the  square  of  the  head  diameter  add  the  square  of 
the  bung  diameter,  and  the  product  of  the  two  diameters  :  multiply 
that  sum  by  ^  of  the  length  of  the  cask  and  by  the  decimal  .0034. 

8.  A  man  bought  a  grindstone  which  was  48  inches  in  diameter 
and  5  inches  in  thickness,  for  $10.  When  he  had  ground  down  3 
inches  of  its  radius,  a  neighbor  proposed  to  purchase  it  from  him  at 
the  same  proportional  price,  in  case  he  would  deduct  4  inches  each 
way  from  the  center ^  allowed  to  be  the  limit  to  which  it  could  be  used. 
What  should  the  purchaser  pay?  Ans.  $7.58-)-. 

Statement  (43''— 8"^). 7854  :  (42''— 8^). 7854  :  :   10  :  Ans. 
Or  (48^'— .8'')  :  (4^^ —8'')   :  :   10  :  Ans. 
Or     66-40  :  50-34  :  :   10  :  Ans. 

oca" 

Or     56-2     :     6«17  :  :  10  :  Ans.=^^. 

112 

9.  If  a  mxin  6  feet  in  height  travel  round  the  earth,  how  muxk 
further  must  his  head  travel  than  his  feet? 

Ans.  37  ^-Q  feet  nearly. 
Let  D=  the  diameter  of  the  eai*th  in  feet ;  then  rti>=  the  cir- 
cumference in  feet.     (D-\-12)=  the  diameter,  and  rti>+13rt=s 
the  circumference  traveled  by  the  man's  head. 
The  difference  =.1^(3.1416)=^««. 


POWERS  AND  ROOm  3ft 

« 

SECTION   UIs 

PO^VERS  ANO  KOOTS. 

The  common  methods  of  operation,  as  taught  under  this  head, 
Rre  in  general  the  besk  One  object  in  this  work  is  to  show  some 
peculiarities  which  will  in  some  instances  abridge  labor,  awaken 
investig'ation,  and  inspire  originality  of  thought. 

We  give  the  following  delinitions  : 

1 .  Any  number  multiplied  into  itself  is  called  the  square  of  that 
number.  Or  we  may  say  (he  product  of  two  equcd  factors  pro- 
duces a  square^     Either  factor  is  called  the  root. 

2.  The  product  of  threie  equal  factors  is  a  cube  or  third  power, — ■ 
of  four  equal  factors,  a  fourth  power,  and  so  on.  One  of  the 
equal  factors  is  a  root  in  all  cases. 

3.  A  square  number  multiplied  by  a  square  number,  will  pro- 
duce a  square  numberv 

N.  B.  This  is  obvious  in  Algebi-a  for  a^  multiplied  by  b^  pro- 
duces a^b^ ,  obviously  a  square,  whatever  numbers  may  be  repre* 
sented  by  a  and  b. 

4.  A  square  number  divided  by  a  square  number  will  give  a 
square  number,  either  whole  or  fractional. 

5.  A  cube  number  multiplied  by  a  cube  number  will  give  a 
cube  number. 

6.  If  a  root  is  a  composite  number,  its  power  (square  or  cube 
as  the  case  may  be)  can  be  separated  in  square  or  cube  factors: 
but  if  the  ix)ot  is  a  prime  number,  the  power  cannot  be  so  sepa^ 
rated. 

We  will  soon  show  the  practical  utility  of  these  principles. 
While  operating  in  powers  and  iHDots  w-e  should  have  the  fol- 


lowing  table  before  us  : 

Numbers,           j    1  j   2  |  3  |   4 

1     5  1     6 

1     7  i      8  1     9 

1     10  1 

Sq.  or  2d  poAver,   |    1  |   4  !   9  jl6 

1   25  1   36 

1   49  1   64  1   81 

!  100  1 

Cube  or  3d  power,!    1  !   ^  !27  j64 

il25  |216 

|343  |512  |729 

iiooo  i 

Powers  being  obtained  from   roots  by  simple  multiplication, 
there  is  no  room  for  much  artifice. 


56  ROBINSON'S  SEQUEL. 

Sometimes  the  application  of  the  following  properties  of  num- 
bers will  be  useful  : 

The  square  of  the  difference  of  two  numbers  is  equal  to  the  sum 
qf  the  squares,  less  twice  the  product  of  the  two  numbers. 
Algebraically,  (a—hy=a'-\-b^—2ab. 
The  square  of  the  sum  of  two  numbers  is  equal  to  the  sum  of  the 
squares  added  to  twice  the  product  of  the  number. 

Algebraically  (a+5)2  =a2 +6^ +2aJ. 

EXAMPLES. 

1.  What  is  the  square  of  79?  Ans.  624L 

(79)2  =  (80—1)2  =6401—160=0241. 

2.  What  is  the  square  of  83?  Ans.  6889, 

(83)2  ^(80+3)2  =64094-480=6889. 

3.  What  is  the  square  of  97?  Ans.  9409, 

(97)2=(100— 3)2  =  10009— 600=9409. 

4.  What  is  the  square  of  971?  Ans.  942841, 

(971)2  =(970+1)2  =940901-f  1940=942841. 

5.  What  is  the  square  of  29?  Ans.  841. 

(29)2  =  (30— 1)2=901— 60=841. 

These  formulas  are  useful  when  one  or  all  of  the  integers  are 
large. 

We  shall  now  turn  our  attention  to  the  extraction  of  square 
root.  We  suppose  the  reader  understands  the  common  method, 
which  as  a  general  operation  is  the  best. 

To  call  out  thought,  however,  we  will  require  the  square  root 
of  9409,  on  the  supposition  that  we  know  nothing  of  the  common 
rule,  and  only  know  thai  two  equal  factors  of  the  numbers  9409  are 
required. 

The  first  thought  is,  that  if  we  divide  any  number  by  any  factor ^ 
the  quotient  will  be  another  factor . 

Take  100  for  one  factor.  Divide  9409  by  100,  and  the  other 
factor  is  94,  omitting  the  decimal ;  but  these  factors  are  not  equal. 
The  factors  sought  then,  or  rather  one  of  them,  is  more  than  94, 
and  less  than  100.  Hence  it  must  be  near  the  half  sum  of  these 
two  numbers  ;  that  is,  near  97. 

By  trial  we  find  97  correct. 


POWERS  AND  ROOTS.  37 

N..  B.  The  half  sum  of  two  unequal  factors,  is  always  a  little 
greater  than  one  of  the  equal  factors,  because  the  sum  of  two  une- 
qual factors  which  form  a  product,  is  always  greater  than  the  sum 
of  two  equal  factors. 

EXAMPLES. 

1.  Find  the  square  root  o/"  841  ;  that  is,  we  demand  two  equal 
f Victors,  which,  multiplied  together,  will  produce  841. 

Assume  any  factor :  say  25. 

25)841(33,  (plus  a  fraction,  which  we  omit)  is 
the  corresponding  factor.  But  these  factors  are  not  equal,  and 
the  equal  factors  must  be  near  their  half  sum  ;  that  is,  near  29. 
By  trial,  29  is  found  to  be  the  number  exactly. 

2.  Find  the  square  root,  or  two  equal  factors  of  the  number 
444889.. 

Divide  by  6.  6)444889^ 

74148 

Here  the  two  factors  scre-veri/  uneqzcal,  but  we  can  bring  them 
to  a  proximate  equality,  by  conceiving  one  multiplied  by  100,  and 
the  other  divided  by  100.  The  factors  will  then  be  600  and  741, 
nearly.  The  half  sum  of  these  is  670,  which  must  be  near  one 
©f  the  equal  factors  sought. 

Now  divide.         670)444889(664 

4020 

4288 
4020 


2689 
2680 


These  factors  being  so  nearly  equal,  and  there  being  a  slight 
remainder,  the  half  sum  of  the  two  (667)  may  be  relied  upon  as 
tlie  true  -root. 


38  ROBINSON'S   SEQUEL. 

3.  Find  the  square  root  of  3.  Am.  1.7320508. 

The  only  two  factors  in  whole  numbers  are  1  and  3  ;*  these  are 
so  unequal  that  their  half  sum,  2,  will  be  entirely  too  large. 
Hence  I  will  assume  one  factor  to  be  1.7. 


1.7)3.    (1.7647 
1.7 

130 
119 


110 


1.7647 
1.7 

2)3.46T7"" 


JQ2  1.7323  root  nearly. 

80 

68 

120 
Making  another  trial  with  the  assumed  factor,  1.732,  we  find 
the  result  as  stated  in  the  answer. 

4.  Find  the  square  root  o/  181.  Ans.  13.45362-f.. 

If  we  allow  ourselves  to  have  some  knowledge  of  square  num- 
bers, we  can  find  a  factor  near  in  value  to  one  of  the  equal  factors 
sought.  Thus  the  square  of  12  is  144,  and  of  13,  169  ;  therefore 
one  of  the  equal  factors  of  181  is  more  than  13.  Assume  it  13.5, 
the  other  factor  is,  then,  13.4074 ;  the  mean  of  these  is  13.4537. 

Taking  this  as  the  assumed  factor,  wo  approximate  still  nearer 
to  the  root  by  a  like  operation  ;  and  thus  we  can  approximate  to 
any  degree  of  accuracy  required. 

By  admitting  that  every  figure  in  a  root  demands  two  places  in  its 
second  power,  we  can  come  near  the  root  at  the  first  assumption. 
For  example  : 

5.  Find  the  square  root  c/ 617796.  Ans.  786. 
Separate  the  power  into  periods,  as  in  the  common  operation ; 

the  superior  period  is  61  ;  the  square  root  of  this  is  near  8,  and 
being  three  periods  the  root  is  near  800.  Assume  780,  then  di- 
vide by  it,,  thus, 

*  In  our  geometrical  problems  we  shall  give  a  scientific  and  satisfactory 
method  of  reducing  unequal  to  the  equivalent  equal  factors. 


POWERS  AND  ROOTS.  39 

780)617796(792 
5460 

7179 
7020 


1596 

1560 


36 
The  half  sum  of  780  and  792,  is  786  ;  the  answer. 

By  the  last  example  we  perceive  that  the  square  root  of  the 
product  of  two  factors  which  are  nearly  equal,  is  very  nearly  equal 
to  the  half  sum  of  the  two  factors.  It  is  a  little  less.  In  the 
last  example  there  was  a  small  remainder,  which  was  rejected ; 
had  there  been  no  remainder,  786  would  have  been  too  great  for 
the  root. 

The  square  root  of  the  producf  of  two  square  factors  is  equal  to  the 
product  of  the  square  root  of  those  factors.  That  is,  the  square 
root  of  a^6^  is  the  square  root  of  a^  into  the  square  root  of  b^; 
in  short  it  is  aX^- 

To  apply  this  principle  I  adduce  the  following  examples  : 

1.  A  section  of  government  land  is  a  square  of  640  acres.  What 
is  the  length,  in  rods,  of  one  of  its  sides  ?  Ans.  320. 

This  problem  requires  the  square  root  of  the  product  of  the  two 
factors,  640  and  160. 

The  product  of  two  factors  is  not  affected  by  multiplying  one  and 
dividing  the  other  by  the  same  number. 

Now  multiply  the  factor  160  by  10,  and  divide  the  other  by  10, 
then  1600  "64  will  be  the  equivalent  factors  ;  both  square  factors  ; 
their  roots  are  40  and  8.     Hence,  the  value  sought  is  40*  8=320. 

Again.  Take  the  original  factors,  640,  160.  Divide  640  by 
2,  and  multiply  160  by  2,  which  gives  320,  320. 

As  the  factors  are  now  equal,  one  of  them  is  the  root  sought. 

2.  A  man  has  50 ^  acres  of  land  in  a.  square  form;  what  is  the 
length  of  one  of  its  sides  ?  Ans.  90  rods. 


Index.       V50|-160  =  7ifi- 160  =  ^405-20=^8100=90. 


40  ROBINSON'S  SEQUEL. 

3.  Find  the  square  root  of  the  product  of  the  two  factors^  1 8 
aw«?32. 

Equivalent  factors,         9  and  64         roots  3-8=24. 
"  Or,  36  and  16         roots  6-4=24. 

Again,      — it —  =25,  which  is  too  great  for  one  of  the  equal 

factors  by  1,  because  the  factors  are  so  unequal. 

In  working  square  root,  it  is  important  that  the  teacher  should 
be  able  to  show  to  his  intelligent  pupils,  that  the  square  on  the 
hypotenuse  of  a  right  angled  triangle  is  equal  to  the  sum  of  the 
squares  on  the  other  two  sides,  notwithstanding  they  have  never 
been  students  in  geometry. 

To  give  an  ocular  demonstration  of  this  important  truth,  we 
present  the  following  figure ; 


The  line  PQ  separates  two  equal  squares.  The  triangle  a  is 
the  right  angled  triangle  in  question,  its  right  angle  at  P,  x  and 
y  are  its  two  sides,  and  the  side  opposite  the  right  angle  P  is 
called  the  hypotenuse.  In  each  square  are  four  equal  right 
angled  triangles.  Let  them  be  taken  away  from  each  square,  and 
in  one  square  the  square  ^will  be  left,  and  in  the  other  square 
the  two  squares  A  and  B  will  be  left. 

Now,  from  each  of  the  two  equal  squares  on  each  side  of  PQy 
we  took  equal  sums — which  must  leave  equal  sums.  That  is, 
A+B=H. 

When  we  operate  on  a  right  angled  triangle,  we  may  divide 
the  two  given  sides  by  the  same  number,  if  we  can  do  so  without 


POWERS   AND  ROOT^S.  41 

a  remainder  on  either  side,  and  then  operate  with  the  quotients 
as  we  would  with  the  original  numbers.  But  in  conclusion  we 
must  multiply  the  result  by  the  number  which  we  divided  bj. 
This  is  working  on  a  similar  reduced  triangle. 

j:xa.mples. 

1.  The  two  sides  of  a  right  angled  triangle  are  312  and  416; 
what  is  the  hypotenuse  ?  Ans.  520. 

Divide  by  52)312.     416 

Divide  by  2)  6  8 

3  4 

Square  3  and  4,  add  those  squares  which  make  25  ;  the  square 
root  of  25  is. 5. 

Multiply         5- 2 -52=520.     Ans. 

2.  A  hawk,  ^perched  on  a  tree  77  feet  high,  was  brought  down  by 
a  sportsman  1 4  rods  distant  on  a  level  with  the  base  ;  what  distance  in 
yards  did  he  shoot?  Ans.  81.15-1-yards. 

14  rods  reduced  to  feet  is  14- 161=7 -.33. 
Now  without  reduction  we  shall  be  obliged  to  square  77  and 
231.     But  we  may  operate  thus, 

7)7-33         77 
11)  33         n 


32-1-12  =  10.  ^10=3.1622+. 

Ans.  in  feet,  =  77  (3.1622).     Ans.  in  yards,  77  (1.054). 


CUBE   ROOT. 


The  object  of  the  Cube  Root  is  to  find  three  equal  factors,  ex- 
actly or  approximately,  Avhose  product  will  give  any  required 
sum.  The  reason  of  its  being  called  cube  is  because  the  three 
factors  may  be  correctly  represented  by  the  length,  breadth, 
and  height  of  a  geometrical  cube.     The  product  of  three  unequal 


42  ROBINSON'S  SEQUEL. 

factors  may  be  represented   by  a  geometrical  solid  of  unequal 
length,  breadth,  and  height,  called  a  parallelopipedon. 

While  operating  for  cube  root  it  is  convenient  to  have  the  cube 
numbers  before  us. 

Roots,      123456789         10 
Cubes,     1         8       27       64     125     216     343     512     729     1000 

We  see  by  these  cubes  that  one  figure  in  a  root  may  Kave 
three  places  in  its  corresponding  power. 

Therefore  separate  the  power  into  periods  of  three  figures  each, 
beginning  at  the  units  ;  the  number  of  periods  will  show  the  num- 
ber of  figures  in  the  root. 

Now  as  we  are  to  have  nothing  to  do  with  the  common  methods 
of  extracting  cube  root,  all  we  are  permitted  to  know  is  the  divi- 
sion of  the  power  into  periods,  and  the  fact  that  three  'equal  factors 
of  the  power  are  required. 

EXAMPLES. 

1.  Extract  the  cube  root  of  84604519  ;  or  in  other  woi'ds,find  three 
equal  factors  whose  product  will  produce  this  number.     Ans.  439. 

Here  are  three  periods  84'604'519,  which  show  that  there 
must  be  three  figures  in  the  root.  The  superior  period  is  84,  and 
84  referred  to  the  line  of  cubes,  its  place  would  be  between  64 
and  125,  whose  roots  are  4  and  5.  Hence  the  root  sought  for  is 
greater  than  400  and  less  than  500  ;  I  should  judge  it  not  far 
from  440.  Therefore  I  assume  440  as  one  of  the  factors  of  the 
number. 

440)84604519(192283 
440 

4060 
3960 


1004 
880 
1245 
880 
"3661 
3520 
"1319 
1320 


CUBE   ROOT.  43 

Now  if  one  factor  is  440,  the  product  of  the  other  two  is  192283, 
very  nearly  ;  (not  exactly,  for  the  last  figure  3  is  too  large  by  a 
very  small  fraction.) 

We  will  now  operate  for   two  equal   factors  of  the   number 
192283,  and  if  our  first  factor  is  near  an  equal  factor,  that  same 
factor  is  near  an  equal  factor  in  this  number  ;  therefore  try  it  thus, 
440)192283(437 
1760 


1628 
1320 


3083 
3080 

Here  we  have  three  factors,  440,  440,  437,  whose  product  will 
give  the  number  84604519,  within  3  units.  These  factors  are 
not  all  equal,  and  of  course  are  not  the  factors  required  ;  but  they 
are  so  nearly  equal  that  one-third  of  their  sum  will  be  one  of  the 
equal  factors  required.     That  is, 

440 

440 

437 
3)1317 

439    Ans. 

2.  Find  the  cube  root,  or  three  equal  factors  of  the  number 
32461759.  Ans.  319. 

By  the  aid  of  the  periods  we  perceive  that  the  factors  must  be 
greater  than  300,  but  nearer  300  than  400. 

Assume  then  312  to  be  near  one  of  the  equal  factors  sought  for. 
Divide  by  312  twice,  or  once  by  the  square  of  312.     That  is, 
97344)32461759(333.3 
292032 


325855 
292032 
"338239 
292032 


46207 
It  is  now  obvious  that  the  product  of  the  three  factors,  312, 
312,  and  333.3,  will  produce  very  nearly  the  given  power;  but 
these  factors  are  not  all  equal,  and  equal  factors  are  required ; 


44  .  ROBINSON'S  SEQUEL. 

but  they  are  so  nearly  equal  that  }  of  their  sum,  319.1,  can  be 
relied  upon  as  extremely  near  the  root  required.  A  factor,  or  root, 
determined  in  this  manner  from  unequal  factors,  will  always  he  a 
little  in  excels  of  the  true  value  required.  Hence,  in  this  case  we 
will  omit  the  one-tenth  and  take  319  as  nearer  the  root  sought, 
and  on  trial  find  it  to  be  the  root  exactly. 

We  will  now  give  one  of  the  most  difficult  examples. 

3.  Find  the  approximate  cube  root  of  16.  Ans.  2.519842. 

The  factors  of  16  are  2- 2- 4;  the  sum  of  these  is  8,  which 
divided  by  3  gives  2.66  for  the  first  approximation  to  equal  factors, 
but  as  these  factors  are  so  unequal,  2.66  must  be  in  excess. 
Therefore  we  assume  2.5  to  be  near  one  of  the  equal  factors  re- 
quired. To  find  the  other  two  factors,  divide  twice  by  2.5,  or 
once  by  6.25. 

Thus,  6.25)16.00(2.56 

12  50 


3500 
3125_^ 
'  /  ~375a 

3750 

Here  we  have  three  factors,  2.5,  2.5,  2.56,  whose  product  will 
give  16  exactly  ;  they  are  not  all  equal  however,  but  being  nearly 
so  \  of  their  sura,  2.52,  is  very  nearly  equal  to  the  root  sought : 
{it  must  he  a  very  little  in  excess). 

Now  if  we  repeat  the  operation  with  2.52  as  an  assumed  factor 
and  find  two  other  corresponding  factors,  ^  of  the  sum  of  the  three 
will  be  the  root  to  a  high  degree  of  approximation. 

4.  Find  an  approximate  cube  root  of  QQ.  Ans.  4.041240. 

By  the  cube  numbers  we  find  that  4  must  be  near  one  of  the 
equal  factors,  therefore  divide  by  the  square  of  4. 
16)66(4.125 
64 
20 
16 
40 
32 


CUBE  ROOT.  46 

Hence  the  root  sought  must  be  a  very  little  less  than  ^  of  the 
sum  of  4,  4,  4.125  ;  that  is,  a  very  little  less  than  4.0416. 

For  a  nearer  approximation  take  4.041  as  one  of  the  factors  of 
66,  and  find  the  other  two,  (fee. 

5.  Jf^ind  an  approximate  cube  root  of  21.  Ans.  2.758923. 

That  is,  find  three  factors,  as  near  equal  as  possible,  whose  pro- 
duct will  be  21,  or  very  nearly  21. 

We  know  that  27  has  three  factors,  each  equal  to  3  ;  therefore 
the  equal  factors  of  21  must  be  each  less  than  3,  and  as  we  can- 
not expect  to  find  the  equal  factors  at  the  first  trial,  we  will 
assume  2.7  and  2.8  to  be  two  of  the  factors,  their  product  is  7.56  ; 
hence,  the  third  corresponding  factor  is  found  by  the  following 
division  : 

7.56)21.00(2.777 
15  12 

5880  '       ' 

5292 


5880 
5292 


5880 
5292 
588 
Here  we  have  three  factors  nearly  equal,  whose  product  is  very 
near  21  ;  one-third  of  their  sum  is  2.759,  which  must  be  a  little 
greater  than  the  root  required.     We  will  therefore  assume  2.75 
as  one  of  the  equal  factors  sought,  and  find  the  other  two  corres- 
ponding factors,  and  one-third  of  their  sum  will  be  an  approxi- 
mate cube  root  of  21. 

It  is  not  necessary  to  give  more  examples. 

When  it  is  necessary  to  multiply  several  numbers  together  and 
extract  the  cube  root  of  their  product,  we  may  often  evade  or 
abridge  the  operation  by  resolving  the  numbers  into  cube  factors. 

EXAMPLES. 

1 .  What  is  the  side  of  a  cubical  mound,  equal  to  one  288  feei 
long,  216  feei  broad,  and  48  feet  high?  Ans.  144. 


46  ROBINSON'S  SEQUEL. 

288=2»12»12 
216=6-   6-   6 
48=4-12 


Product,         288»216«48=123>63«8 

Whence,     y288- 216 -48=12*6*2=144.     Am, 

S.  Required  t^ie  cube  root  of  the  product  of  448  by  392  in  a  brief 
manner. 

N.  B.     Divide  by  the  cube  number  8  ;  then  it  will  appear  that 
448=:8'8-7 
and     392=s8-7»7 


Product,  448-392=^83-73 

Whence,         3^448*392^8- 7=66       Ans. 

3.  Find  the  cube  root  of  the  pn)duct  of  the  two  fuctors  192  and 
1025  in  as  brief  a  manner  as  possible,  Ans.  60. 

The  three  last  examples  are  rare  cases  ;  nevertheless  they  serve 
to  awaken  thought,  afid  for  this  purpose  they  were  introduced. 


JlI.I.I«ATIOM   AI>TE»]VATE. 

No  arithmetical  rule  is  more  difficult  to  be  comprehended  by 
young  pupils  than  this. 

The  operations  are  generally  very  trifling,  but  the  rationale  is 
rarely  discovered. 

For  this  reason  we  shall  be  a  little  unique  in  our  exposition  of 
the  principle — we  shall  resort  to  an  experiment  in  philosophy. 

It  is  clear  to  the  comprehension  of  almost  every  one,  that  two 
bodies  balanced  on  a  fulcrum,  the  heavier  body  must  be  nearer 
the  fulcrum  than  the  lighter  body. 

Thus  two    bodies  bal- 


anced on  the  fulcrum  F, 
2  pounds  at  the  distance 
of  6,  will  balance  6  pounds  at  the  distance  of  2. 


ALLIGATION  ALTERNATE.  47 

Or  when  we  have  the  distances,  we  can  take  those  distances, 
or  their  proportion  for  corresponding  weights  if  we  alternate  them. 
That  is,  the  long  distance  must  go  on  the  opposite  side  of  the 
fulcrum,  and  there  become  weight,  and  so  of  the  other  distance, 
and  there  will  be  a  balance. 

We  shall  apply  this  principle  in  the  following  example. 

1.  A  grocer  has  two  kinds  of  sugar ,  one  at  9  cts.,  the  other  at  16 
cts.  per  pound  ;  he  wishes  to  make  a  mixture  worth  1 1  cts.  per  pound : 
what  proportion  of  the  two  kinds  shall  he  take  ? 

Here  two  quantities  are  to  be  balanced  on  tho,  fulcrum  11. 
The  difference  between  9  and  11  is  2  ;  place  C   95 

the  2  opposite  16.     The  difference  between  11  and        11  ^ 
16  is  5  ;  place  this  opposite  9.  (^16     2 

The  result  is,  that  5  pounds  at  9  cts.  =  45  cts. 
and  2  pounds  at  16  cts.  =  32  cts. 
Makes  7  pounds  worth  77  cts.,  which  is  11  cte. 
per  pound  as  required. 

We  may  now  expand  the  problem  and  add  another  kind  of 
sugar,  worth  10  cts.  per  pound. 

Then  make  a  mixture,  worth  11  cts.,  with  sugars  worth  9,  10, 
and  16  cts.  per  pound. 


9\   5 
16^  2+1 


Link  each  price  below  1 1  to  the  one  above. 
Make  a  balance  between  9  and  16  as  before, 
then  between  10  and  16,  and  all  will  be  bal- 
anced as  required.  The  result  is.  6  pounds  at 
9,  5  at  10,  and  3  pounds  at  16  cts.  That  is,  13  pounds  of  this 
mixture  is  worth  143  cts.,  which  is  11  cts.  per  pound  as  required. 

On  the  same  principle,  any  number  of  ingredients  may  be  re- 
duced to  any  given  mean  price  or  quality. 

We  give  but  one  example. 

Mix  6  bushels  of  oats,  worth  20  cts.  per  bushel,  with  8  bushels  of 
oats  worth  25  cts.  per  bushel,  with  rye  at  70  cts.  per  bushel,  and 
wheat  at  80  cts.,  and  sell  the  mixture  at  75  cts.  per  bushel ;  what 
proportion  of  rye  and  wheat  will  there  be  in  the  mixture  ? 

Ans.  Rye  14  bushels,  wheat  160' bushels. 

6  bushels  at  20  cts.  will  cost  120  cts.  and  8  at  25  cts.  will  cost 


48  ROBINSON'S  SEQUEL. 

200  cts.;  whence  the  14  bushels  of  oats  will  cost  320  cts.,  or  22IJ 
cents  per  bushel. 


76 


22f    5 
80^  6+621 


6    bushels  of  oats. 
6    bushels  of  rye. 


51}  bushels  of  wheat. 
Here  we  have  a  true  mixture  worth  76  cts.  per  bushel,  but  the 
mixture  contains  only  6  bushels  of  oats  :  it  must  contain  14,  there- 
fore multiply  each  of  these  quantities  by  y.     Then  6*  Y  =  14. 
57}- V  =  160. 

Alligation  is  of  little  or  no  practical  utility,  yet  it  serves  as  well 
as  any  other  arithmetical  operation  to  discipline  the  mind. 


POSITION. 

SINGLE    POSITION DOUBLE    POSITION. 

Before  Algebra  became  a  popular  study  many  algebraic  prob- 
lems appeared  in  common  Arithmetics,  and  were  solved  by  special 
rules,  which  were  drawn  from  the  results  of  algebraic  investiga- 
tions. But  at  the  present  day  all  such  problems  in  Arithmetic 
are  improper ;  as  much  so  as  to  travel  500  rdiles  in  a  pri- 
vate carriage  by  the  side  of  a  railroad  track. 

Problems  in  Single  Position  produce  equations  reduceable  to 
this  form:  a:f—m.  (1) 

Problems  in  Double  Position  produce  equations  in  this  form  : 
ax-\-bz=m  (2) 

Not  knowing  the  value  of  x  in  equation  (1)  we  assume  some 
known  number,  x,  which  may  not  be  the  true  one,  and  if  it  is  not, 
the  result  will  not  be  the  given  number  m  ;  let  it  be  m\  Then  we 
shall  have :  ax'=m'  (3) 

Divide  equation  (1)  by  (3),  then 
X  —-.m  ^ 
X      m' 
Converting  this  into  a  proportion,  we  have 
m    :  m  '.  '.  x'  '.  V, 


POSITION.  40 

The  result  of  this  proportion,  put  into  words,  is  the  rule  of 
Single  Position  given  in  all  the  old  Arithmetics. 

Rule.  Assume  a  number  and  find  the  result  of  the  supposition  ; 
then  say  :  As  the  result  of  the  supposition  is  to  the  given  result,  so  is 
the  supposed  number  to  the  true  number. 

We  give  but  a  single  example. 

A  and  B  have  the  same  income,  ^contracts  an  annual  debt 
amounting  to  |  of  it :  B  lives  on  |  of  his  income,  and  at  the  end 
of  10  years  lends  to  A  money  enough  to  pay  off  his  debts  and  has 
$160  to  spare  :  what  is  the  income  of  each  ?  Ans.  $280. 

For  the  sake  of  convenience  we  will  take  some  number  divisible 
by  6  and  7  ;  therefore  take  35  for  the  supposed  income  of  each. 

Then  ^'s  debt  in  one  year  is  $5,  in  10  years  $60. 

B  saves  ^,  or  $7,  in  one  year,  in  10  years  $70. 

B  lends  A  60  and  has  $20  left  as  the  result  of  the  supposition. 
Then,  20  :  160  :  :  35  :  280.     Ans. 

Now  let  us  suppose  the  income  to  be  1,  or  unity.  Then  ^*s 
debts  in  10  years  amount  to  y . 

B  saves  in  10  years -^j",  or  2. 

B  pays  A's  debts  ;  he  then  has  (2 — y>)=^. 

Whence,         4=160,     or4=40,     or  1=280.     Ans. 

This  manner  of  working  by  fractions  some  teachers  call  Arith- 
metic, but  it  is  Algebra  in  disguise. 

Let  X  be  the  income,  in  place  of  1,  and  the  identity  will  be 
obvious. 

To  show  the  arithmetical  rule  for  Double  Position  we  take  the 
equation 

ax-\-b=m.  (1) 

1st.  Suppose  a;  to  be  represented  by  the  assumed  number  x', 
and  m-\-e'  the  result  of  this  supposition,  e'  being  the  excess,  or 
error.     Then, 

ax'+b^^m+e'.  (2) 

Again,  assume  another  number,  say  x"  and  e"  the  second  error. 
ax"+i=m-{-e".  (3) 

Subtract  (1)  from  (2)  and 

a  (x' — x)=e'.  (4) 

(1)  from  (3)  alx'''-<c)=e\  (6) 

4 


60  ROBINSON'S  SEQUEL. 

Divide  (4)  by  (5)  and  we  have, 

x' — X      e' 


X  — X       e 
Whence,  e"x — e"x^e'x" — e'x. 

ex" — e"x' 


And  a;=- 


This  last  equation,  put  in  words,  is  the  rule  given  in  all  the 
Arithmetics  of  a  former  day.     It  is  substantially  this  : 

Make  two  distinct  suppositions  and  note  the  results.  Take  the 
diflference  between  the  given  result  and  the  result  of  each  suppo- 
sition, which  difference  call  error.      Then, 

Multiply  the  first  error  hy  the  last  supposition,  and  the  last  error 
hy  the  first  supposition.  Divide  the  difference*  of  these  products  hy 
the  difference  of  the  errors,  and  the  quotient  will  he  the  number 
required. 

EXAMPLE. 

A  has  820;  B  ha^  as  many  as  A  and  half  as  many  as  C;  and 
C  has  as  many  as  A  and  B  both.     How  many  dollars  had  ea^h? 

Ans.  A  820;  B  860;  C  880. 
Suppose  C  had  60. 
Then       B  had  30+20=50. 

A  had  20.     But  20+50  is  not  60,  the  error 
therefore  is  10. 

1st  sup.  60.     Error  10.         660 


2d  sup.  m.     Error    7.         420 

3)240 

Ans.         80 


Again,  suppose  Chad  QQ. 
Then  B  had  33+20=53. 
A  had  20. 
But  20+53  is  not  66.    Error  7. 
By  Algebra, 

Let  2a:=  C's.     Then  a;+20=^'s.     20=^'5. 

Then  ^x=x-\-^0.        ic=40.     2x=i^0.Ans. 
By  comparing  the  last  operation  with  the  operation  of  the  arith- 
metical rule,  and  then  applying  it,  we  perceive  the  folly  of  retain- 
ing the  old  rules  for  mere  arithmetical  purposes. 

The  rule  of  Double  Position,  however,  is  of   importance  in 
solving  exponential  equations  in  Algebra. 

*  The  diflference  is  Algebraic  dz  ;  hence  some  Arithmetics  give  two  cases 
to  the  rule — Qne  when  the  errors  are  alike,  the  other  when  unlike. 


PART  SECOND 

AI.O£BRA. 

SECTION   I, 

In  Algebra,  as  in  Arithmetic,  we  shall  only  touch  here  and 
there  on  such  points  as  might  come  up  in  the  school-room,  and 
present  some  difficulties.  Hence  this  work  will  seem  to  want 
connection. 

When  we  indicate  the  solution  of  a  problem  in  Robinson's 
Algebra,  University  edition,  we  shall  refer  to  it  by  Article  and 
number  of  the  problem,  and  not  write  the  problem  in  full. 

When  the  problem  is  to  be  found  only  in  some  other  book,  or 
is  original  here,  it  will  be  written  out  in  full. 

ROBINSON'S    ALGEBRA, 

SECTION   II. 

CHAPTER  I. 

EQUATIONS. 

None  of  the  questions  in  this  chapter  require  the  aid  of  a  key, 
«intil  we  come  to  the  15th,  page  65. 

fl5.)         (^J_::±L-^Y=='1^1^-J^=  his  stock  at  the 
\      3  /3  9  3 

'commencement  of  the  third  year,  before  his  expenses  are  taken 

out.     Hence, 

Reduced  gives  a-=  14800,  Ans,  ' 

(16.)  Put  a=99,  .'r=time  past.  Then  v. — :rc=time  to  come, 
and  per  question, 

-^  _= ...,.,.  ,a's=54. 

3  6 

(17.)  Let  ^=  the  whole  composition. 
Then  per  question, 


H  ROBINSON'S  SEQUEL. 

_+10=nitre. 

- — 4|^=  sulphur. 

— 4- — — 2= charcoal. 
21^7 

By  addition,       f^-j-^-j-^f +31+12 =ar.. 

Multiply  by  6,  and  drop  5x  from  both  sides,  and  we  have 
l^+21+^=a;.      Or,    4a:+21  -7+60=7^:. . .  .rr=69. 

(18.)  Put  a=  183;  a:=what  the  first  received;  then  a — ^ar= 
2d  received. 

Per  question,  ^^Sa—3x ^^^^^ 

^  7  10 

(19.)  Put  a=68,  x=  the  greater  part,  and  a — ar=  the  less. 
84— <r=3(40— a+a;) a:=42, 

(20.)  The  distance  from  ^  to  ^  put  =2x. 
The  distance  from  C  to  D  *'    =3a*.  . 

Then,  3  times  the  distance  from  B  to  C  must  be 

-+ —  or  the  distance  is,  -+-. 
2^  2  6*2 

Hence  the  whole  distance  is,  5a:+-+-=34.. 

6     2 

(21.)  Letar=fhe  flock. 

The  first  party  left  him  ?f — 6.    The  second  left  -—3—10=2 
^     "^  3  3  • 

(23.)  Observe  that  for  every  vessel  he  broke  he  lost  12  cents: 
S  cents  fee  and  9  cents  forfeiture. 

300— .I2a;=240 x=5. 

(24.)  Had  he  not  been  idle  he  would  have  been  entitled  to  o^ 
cents.     But  he  was  idle  x  days  at  a  loss  of  (i+c)  cents. 

Hence,  ab — (b'^c)x=sd,  x= . 


ALGEBRJL.  63 

{"25.)  Put  6a;=  less  part.     Then  a — 5«=  the  greater  part. 

3 
Per  question,  a — 7a; = 20a; — _  (a — 6a;) 

7a—49x==  1 40a;— 3a+16a; 
or,         204a;=10a=10-204 
or,         a;=10 
Therefore,  5a;=60=  the  less  part 

(26.)  Let  8a;=the  price  of  the  horse. 
Then  a — 8a;=:  the  chaise.     «3=34i. 

Per  question,       2a — 16a; — 3a;=24a; — -(a — 8a;) 

5 
or,       2a = 43a; — -(a — 8a;) 

14a=301a;— 6a+40a; 
19a=341a;         or,         a;=19 

8a;=152.  Ans, 

(29.)  Let  5x=  his  money. 

After  he  first  lost  and  won  45.,  he  had  4a;-|-4. 
He  again  lost  and  won,  and  then  had  3a;-|-3-{-3. 
I  of  this  must  equal  20,     or,     3a;-l-6=24.        x=6 

5a;=30.  Ans. 

(30.)  Let  3a;=  the  income. 
Then  2a;=  the  family  support. 

a;— ?^  =^=70.     Hence, .  - . .  3a'=70  •  9. 
3      3 

31,  32,  and  33  require  no  explanation. 
(36.)  Last  year  the  rent  was  x  dollars. 

This  year  it  is       x4-^=  1 890. 
^  ^100 

(36.)  Is  the  (35)  in  g-eneral  terms. 


(37.)  Let  7a;=  equal  the  income. 

5 


7a; 
Then  x=  A's  annual  debt.         — =what  B  saves. 


7a; 

—— <c=16  or        a;=40.  7a;=280.  Ans. 

5 


54  ROBINSOlSr^S  SEQUEL. 

In  general  terms, 

6  2  4  . 

(38.)  f^4-^4.?^=a 

(39.)  Let  10aj=  the  income. 

Then           2a;-|-100=  the  sinn  spent. 
Sar-f-  35=    "    sum  left. 
7^+135=  1  Oa:  the  whole, 
or,  45=ar 450.  Ans. 

(40.)  Let  2lx=  the  income. 

Then  3x-\-a=:  the  sum  spent. 

7x-\-b  =  the  sum  left. 
10ar-f-«+^=21a:=  the  whole.       ^^(a-f-^) 


11 


(41.)  2a;4.4     :     3a:-f4     :   :     5     i     7. 

(42.)  Let  x^ — 7=  the  number. 

Then,  per  conditions,         x — \=.Jx^ — 7 
a;2_2ir-f-l=:a:2— 7 
or,  2:= 4     and     x^ — 7=^. 

(43.)  ^'s  rate  of  travel  is  \  miles  per  hour. 
J5's  rate  of  travel  is  f  miles  per  hour. 
A  is  in  advance  when  B  sets  out,  ^f  mdks. 
Let  ar=  the  hours  after  B  starts. 

Then^  — = — + —     Reduced  gives a;=42. 

3       5       5 


CHAPTER  II. 
EQUATIONS- IN   TWO  UNKNOWN   QUANTITIES. 

(6.)  Add  the  two  equations  together,  representing  {x-^y)  by  4, 

and  we  have        is+^5=50  or 5=5  •3. 

But  ir+9y=21-3 

Subtract        x-\-  y=  5-3 

8y=16-3         or, y=e. 


ALGEBRA.  ti 

(7.)  B^  adding  the  two  equations  we  have 

65=50 
or,  a:-|-y=10 
but  4a-+y=34 
Hence     3x       =24       or, ar=8. 

(9)  and  (10)  are  resolved  same  as  (6)  and  (7). 

(11.)  From  the  first  equation  we  have 

y=2:r— 80. 
Transpose  — 8  in  the  second  equation  and  we  have 

6    ^3         4     ^ 
Multiply  by  60  and  we  have 

1 2a;+l  22/+20a:=30y— 1 52r+35  •  60 
or      47a;=18y4-35-60 
Substituting  the  value  of  1  By,  we  have 
47a;=36.r— .18-80+35-60 
or     lla;=— 240-6+350-6=110-6 
Hence, x=60. 

(14.)  Bringing  unknown  terms  to  the  first  members  of  the 
equations  and  we  have 


x=4. 


^_^— — 1 

4 

_2 

_3 

X     y 

y 

X 

2 

By  addition.    -=-  or 

J 

x     2 

(15.)  Putc 

f=50. 

Then, 

a:+3a     : 

y—  a     : 

3 

:     2 

And 

X —  a     : 

y+^a     : 

: 

5 

:     9 

2a:+6a 

=3y— -  3a 

(1) 

^x — 9a 

=53^+10a 

(2) 

Multiply  (1)  by  5,  and  (2)  by  3  ;  then, 

10:P+30a=15y— 15a  (3) 

27a:— 27a=  1 5y-[-30a  (4) 

Subtract  (3)  from  (4)  and 

17a: — 57a=45a 
17a;=102a 

ar=6a=6  •  50^300. 


M  ROBINSON'S   SEQUEL. 

(16.)  Divide  the  numerator  of  the  second  member  of  the  first 
equation  by  its  denominator,  and  we  have 

Hence,  Ux-\-Uy=m  (1) 

Multiply  the  second  equation  by  (3y — 4)  and  we  shall  have 

9^_12.=(i51=?^^LPj^^^^  10. 

or,      no^n.=(l^ti}M(^tz^- 

4y— 1 
440y_48a^— 1 1 0+1 2a:= 453y— 48a:y— 604+64a;. 
0=13y+52a:— 494 
or,  4a;+y=38  (2) 

Add  (1)  and  (2),  and  we  have 

15(x+y)=165 
or,                          a;+y=ll                        (3) 
(3)  from  (2)  gives         3a;=27 x=9. 

(17.)  Multiply  the  1st  equation  by  14  and  we  have 
42a;— 7y=49 
Add  — <c+7y=33 


41a;         =82        or, 

x=2. 

(18.)                              ^^2^^i6 

a;-^=-3 
6 

Subtract              ?y-|-?y=19 
3       5 

10y+9y=19-16 y-. 

=  16. 

(19.)  Divide  2d  by  the  1st,  and                 a;— y=2. 
But                                          a:-|-y=8. 

(20.)  Multiply  the  first  equation  by  (X'\-y),  and  the  second 
by  9,  and  we  have 

4(a;+y)2=9(a:2— y«)         9(x^-^y^)=9'36. 
4{x+yy=9'36. 
Hence,           x-^y=9. 
Divide  the  first  equation  by  this  last,  and  we  have    a?— y=4. 

ALGEBRA. 


(21.) 


_4y 


64y3 

'  27 


27       ^ 
372/3=37-27 y=3. 

(22)  and  (23)  require  no  remarks. 

(24.)  The  first  equation  gives 

a;+24y=  91  (1) 

Add         40a:-l-y=763  (^) 

Multiply  (1)  by  40,  and  subtract  equation  (2),  and 

969y=2877     or y=3, 

(26.)  From  1st  equation  take  the  2d,  and  we  have 

2ia;+5y=60. 
Divide  by  2^  and  we  have        x-\-2f/=24: 
But  1^+2^=19 

^x         =5      or,  ...a;=10, 

(26.)  Add  the  two  equations,  and 

Y(^+y)+K^+y)+2o=^+y 

or,  -+-+20=s .5=120. 

By  2d  equation,  1(120) — 5=y=35.     Ans. 


(7.)  Given 


CHAPTER  III. 
EQUATION'S  OF  THREE. OR  MORE  UNKNOWN  QUANTITIES. 

Sy=u-\-x-\-z 
Az:=u-\-x-\-y 
u^=x — 14 
Subtract  2d  from  the  1st,  and 

2a; — ^y=y — x  or 

Subtract  2d  from  the  3d,  and 

42 — 3y=y — z  or 

Add  3d  and  4th  and 


to  find  u,  X,  y  and  z. 


3x=4y     (1) 


5z=4y     (2) 
4z=2x-{-2/—'14  (3) 


■^' 


58 


ROBINSON'S  SEQUEL. 


Multiply  (3)  by  5  and  (2)  by  4  and 

202=10a:+  5y— 70 

202= \6y 

0=10a;— lly— 70 
'    <»  30x— 33y=210 

From  equation  ( 1 )        30a: — 40y=     0 

7y=210 

y=  30 


(X 


(8.) 


+1+^=62 
'3^4 


I  3^4^5  ^ 

4^5^6 


b 


To  avoid  numerical  multiplication,  and  really  to  understand 


Igehra  as  applied  here,  observe  that 

62+38=100 

Put  a=60;  then     62=a+12=a+5. 

Clearing  of  fractions,  we  have 

QxJ^  4y+  32=12a+126 

(1) 

20a?+l  5y+ 1 22=60a— 1 66 

(2) 

\5,x-\-ny-\-\0z=ma—QQb 

(3) 

Multiply  (1)  by  4,  and  subtract  (2). 

Then,         4a;+y=636— 12a 

(4) 

Subtract  (3)  from  (2),  and 

5ar+3y+22=  456 
3 

(5) 


15a;+9y+63=1356 
Subtract  \2x-{-^y\-Qz=  Mb-\-Ma 

~3ar+  y         ^TTT6— 24a 
Subtract  (5)  from  (4),  and  we  have 

a:=(12a— 486)^12(a— 46)=12-2.  Ans. 
That  is,  a:=24     or     26. 

Now,  equa,tion  (4)  gives  us 

864-y=63J— 6a 

y=(55—a)6=6- 12=60. 

(9.)  By  adding  the  three  equations  and  reducing,  we  have 
4a;+3y+22=3a  (1) 


ALGEBRA.  50 

By  adding  the  2d  and  3d,  reducing  and  doubling,  we  have 
l0x-{-4y+2z=4a  (2) 

Subtracting  (1)  from  (2),  and  we  have 

6x+y=a  (3) 

Adding  the  1st  and  3d,  and  reducing,  we  have 

x-\-2y=a  (4) 

From  (3)  and  (4)  we  readily  find  x  and  y. 

'2x  +y  —2z  =40  (1) 

4y  —X  +32  =35  (2) 

(10.)  <  Su  +t  =13  (3) 

y  J^u  +i=15  (4) 

Sx—y-\-St—-u=49  ( 5) 

It  is  easier  to  eliminate  t  than  any  other  letter. 

Subtract  (3)  from  (4),  and  we  have 

y—2u=2  (6)  ^ 

Three  times  (3)  taken  from  (5),  gives 

.Sx—y—10u=10  (7) 

Add  (6)  and  (7)  and  divide  the  sum  by  3,  and 

x—4u=4  (8) 

Double  (6),  and  subtract  it  from  (8),  and  we  have 

x=2y  (9) 

Eliminate  z  from  equationi^  (1)  and  (2),  and  we  have 
4a;+lly=190 

But4x=Sy.      Then  1%=  190,       or y=10. 


PROBLEMS  PRODUCING  SIMPLE  EQUATIONS,  INVOLVING  TWO 
OR  MORE  UNKNOWN  QUANTITIES. 

(1.)  Let  X,  y,  and  z  represent  the  numbers. 

xy=600                       (1) 
xz=SOO                      (2) 
yz=200                      (3) 
Multiply  (1)  and  (2),  and  divide  the  product  by  equation  3, 
and  we  have  x^  =900 a:=30. 


60  ROBINSON'S  SEQUEL. 

(2.)  Let  X,  y,  and  z  represent  the  numbers.     Then  per  question 

y+fZ:!=  70  (2) 

o 

^+y+^=190                      (3) 
Double  (1)  and  subtract  (3),  gives a;=50. 

This  problem  calls  the  pupil's  judgment  into  exercise.  He  does 
not  know  in  the  first  place  which  is  greater,  x  or  z',  hence  he 
must  try  both  suppositions,  and  the  one  that  verifies  equation  (2) 
is  right. 

(3.)  Let  a;,  y,  and  z  represent  the  shares,  and  put  a=120 

^ — T(y+^)=« 

y—({x+z)=a 

Clearing  of  fractions,  we  have 

Ix — 4y — 42= 7a  (1) 

_-3;y_(_8y— 32=8a  (2) 

— 2a;— 2y+92=9a  (3) 

Double  (1),  and  to  the  product  add  (2),  and  we  have 
liar— ll2=22a 

a;—     z=  2a  (4) 

Double  (3),  and  to  the  product  add  (1),  and  we  have 
— lla;+222=lla 

—x-\-  2z—     a  (6) 

Add  (4)  and  (5),  and  we  have g=3a=360. 

Another  solution. 
Assume,  x-\'y-\-z=s. 

By  the  question        {/-^(/+^)=- 
as  before,  i  y— f(^+^)=« 

Clearing  of  fractions,  and  we  have 

7a;_4y— 42=7a  (1) 

8y— 3a;— 32= 8a  (2) 

90— 2a:— 2y=9a  (3) 


ALGEBRA.  » 

To  ix-\-4y-\-4z=::4s 

Add  (1)  7ar — 4y — 42= 7a 

And         ll^  ^4s+7a         (4) 

To  3x-{-Sy-{-Sz=3s 

Add  (2)  Sy—3x—3z=Sa 

And        Tly  ^35-{-8a         (6) 

To  2a:+2y+22=25 

Add  (3)  90— 2a;— 2y=9a 

And         ~Uz  ^28-\-9a         (6) 

Add  (4),  (6)  and  (6),  observing  that  {x-^-y-\-z)=s,  and 
ll5=95+24a 
Whence,  «=12a 

This  value  of*,  put  in  (4),  gives 

llx—4Sa-^7a=55a 
or  a?=:5a=600.   Ans. 

(4.)  Resolved  in  the  book. 

Let  X,  y,  u,  and  z  rejpresent  their  ages,  and  8  their  sum. 

Then,  «— 2=18 

5 — «=16 

^ — y^=\4 

8—x—\2 


By  addition,  35     =60 s=20. 

(6.)  Let  x^=A*&  shillings.        > 

y=B*s 

z—C'a       " 

After  the  first  game  they  will  have  as  here  represented 
X — y — z=A 
2y  =B 

2z  =0    ' 

After  the  second  game, 

2ar— 2y— 22=^ 
3y—  .t—  z—B 
4z  =(7 


n  ROBINSON'S   SEQUEL. 


After  the  third  game 

4x—4y—4z==   16 

(!) 

6y— 22;->22=   16 

(2) 

70—  ar—  y==   16 

(3) 

Sum,                       a:+y-{-  2=3*16 

(4) 

Add  (3)  and  (4)  and  we  have 

82=4-16 2=8. 

(6.)  This  problem  is  resolved  in  the  book,  by  equation  7, 
Art.  53. 

(7.)  Let  X  represent  the  better  horse,  and  y  the  poorer 

a:+15=  Uy+10) 

x+10=.my+15) 
Therefore,            |(y+10)=||(y+15)+5 
Reduced  gives .,, y=s50. 

(8.)  Let  x=  the  price  of  the  sherry. 

y= brandy. 

Put      a=78. 

2x-{-  y=3a 
7x-\-2y=9'a-\-9 
'3x         =3a-f9 

X         =  a-\S— 81s.  Am, 

(9^,)  Let  x=:A's  time.        y=^B's  time. 

Then,  _=th-e  part  that  B  can  do  in  one  day, 

4  ,  4_^^1 
xy     16     4- 


Hence , ys=48. 


36^3 
J     4 

(10.)                  2ar  _2  f+?=^ 

^+7     3  '^f     5 

Sx=y-\-7  5a:+10=6y. 

(11.)                    ,  2y  ,  3ar 

^3  ^4 

(12.)  Let  .r=  the  greater,  and  (24 — x)=  the  less, 

X         ,     24 — X  ..4.1 


24 — ^  X 


ALGEBRA.  63 

x^     :     ^24— a:)2     :  :     4     :     1 
By  evolution,         x     :     24 — x     :  :     2     :     1 
(13.)  Let  a:=  the  number  of  persons. 
y=  what  each  had  to  pay. 
Then,      xy=:  the  amount  of  the  bill. 

Put        (a;+4)  (y— 1)= the  bill. 

Also-,      (x—3)  (y+l)= the  bill. 

xy-\-47/ — X — 4=xy 
xy—Sy-{-x—S=:xy 

4y — X — 4=0 

— 3y-^x — 3=0 

By  addition,  y      — 7=0 

(14.)  10x-\-y=4x-\-4y         or, y=:2x» 

10x-{'y^27=10y+x 
'  or,  9x      +27=  9y 

X      +  3=     y=^2x,  hence,      «=3. 

(16.)  Let  x=  the  digits  in  the  place  of  lOO's. 
y=  *'       in  the  place  of     lO's. 

z=  "  the  units. 

W-^y-\'Z=\l  Z=:z2x 

100ic-f-10y+2+297=  lOO^+lOy-f-a? 

99a;+297=  99^ 
x-\-3=z=:2x  Hence x=3. 

/ic  \  T   +     a;— 40       X — 20  ,     a:— 10  .  ^, 

(16.)  Let ,     .,     and     _ — represent  the  parts. 

Then,  f=12+f:r?2+5ld?=90 .=100. 

2      ^     3     ^     4 

(17.)  Let  X  represent  the  part  at  5  per  cent,  and  (a — x)  the 
part  at  4  per  cent.     Then 

5x  1  4a — 4x , 

ioo"*"    100 
Hence x=:100b — 4a. 

(18.)  To  avoid  high  numerals,  and  of  course  a  tedious  opera* 
tion,         Put  a= 5000;     then  2a=  10000,     3a=s  16000, 

—=1500,  and    A^^=800. 
■  10  100 


64  ROBINSON'S  SEQUEL. 

Put              x=.  A'&  capital,  and  r — l=^'s  rate. 
ar+2a=  ^'s      **  r=B'&  rate. 

a;-j-3a=  (7's      "               r+l==C"s  rate. 
''  r  X — X  I  16a r  x-\-2a  r 


By  conditions 


100     '  100        100 

rx — X   .3a_rx-{-3ar-\-x-\-Sa 

~ioo  ~^io  Too 

Reducing  (1),  gives  a:=(16 — 2r)a 


(1) 

(2) 


Hence, 


32— 4r=27-~3r, 


or. 


a 
. .  .  .r=5. 


(19.)  Put  a=1000,  X  and  y  to  represent  the  two  parts,  and  r 
and  t  the  rates  expressed  in  decimals. 
'  x-\-y=z\3a 
rar=    ty 
te=360 
ry=490 
Divide  (3)  by  (4),  and  we  have 

/J\/a;\     36 
W\y/     49 


Then  by  conditions, 


(1) 

(2) 

(3) 
(4) 

(6) 


From  (2)  we  have 


x_t 
y  r 
t  . 


Substitute  the  value  of  —  in  equation  (5),  and 


.36 
49 


or 


By  returning  to  equation  (1)  we  have  _iL-|-y=13a. 

13y=13a*7         or y=7a. 

(20.)  Let  Xy  y,  and  z,  represent  their  respective  ages. 
Then  by  conditions  given,         x — y=.     z 
5y-{-2z — ^a;=147 

(21 .)  Let  X,  y,  and  z,  represent  the  respective  property  of  each, 
and  put  5=  their  sum. 


ALGEBRA.  66 

fX'\-3y-\-3z=i7a  a=100. 

Conditions,  }y-\-4x-\-4z=58a 

(z-^5x-\-5y=63a 

Add  2x  to  the  1st  equation,  3y  to  the  2d,  and  4z  to  the  3d, 

observing  that  x-\'y-\-z=s  ;  then  we  shall  have 

3s=47a+2a;  (1) 

45=58a+3y  .  (2) 

65=63a+42  (3) 

3s— 47a 

.     or,  x= 

2 

45 — 58a 

5s— 63a 
z=. 

4 

3*— 47a  ,  4s — 58a  ,  bs — 63a 


By  addition,  s 


2        '        3        '        4 
Hence, 5=19a. 

This  value  of  s,  put  in  equation  (1),  gives  a?=5a=500. 


(23.)  Let  X,  y,  and  z  represent  the  respective  sums. 

.+^=<. 

(1) 

y+|=a 

(2) 

1    ^ 

.   (3) 

ac+y=2a 
3y+2=3a 
42+ar=:4a 

From  the  1st 

or,         4z-\-12y= 

or,         4z-f-    x= 

—x-\-12y= 

24x-\-12y= 

=  12a 
:  4a 
:  8a 
:24a 

25x  =16a 


(24.)  This  problem  is  resolved  in  the  work,  by  the  13th  exam- 
ple, page  80,  (Art.  61.) 
5 


6  ROBINSON'S  SEQUEL.  f 

(25.)  Let  aj-sthe  greater,  and  y  the  less. 

^x — y=0        or, ...2ir=3y. 

(26.)  a:+Ky+^)=«=61 

2y+(^+2/+2)=3a 
324-(^+y+2) = 4a 
a;=2a— 5  (1) 

y=i(3a— 5)  (2) 

2=H^a-s) (3) 

s=  2a—  5-|-i(3a— s)+^(a— s) 
6s=12a— 6«-|-9a — 35-f-2a— 25 

175=29a     or .5=29-3=87. 

Ifow  equations  (1),  (2),  and  (3),  will  readily  give  x,  y  and  z. 

(27.)  Let  x=A'&,  y  JB's,  and  z=C's  sheep. 
Then  by  the  conditions, 

a:_|_8— 4=y-[-g— 8 
i(y+S)=x+z-S 
i(z+S)^x+y-S 
x+n=  y+  z  (1) 

y^24=2x-\-^z  (2) 

2-|-32=3a;+3y  (3) 

Add  (1)  and  (3),  and  we  have  x-\-44=3x-\-^. 

Double  (1),  and  subtract  (2),  and  we  have 

2x — y=2y — 2x  or,  4x=3i/ 


y=8. 


(30.)  This  is  a  repetition  of  the  1 0th  example,  page  89,  in- 
serted here  by  oversight. 


But 

2a;+4y=44 

4a;-f-8y=44-2 

ny=44-2         or 

(28.) 

ar+l_l                                X    __1 

y      3                      y+1    4 

(29.) 

«+2_6                                X    _1 

y        7                            y+2    3 

ALGEBRA.  ♦      ^ 

(31)  Let  a?i=:-4's  money,  and  y=zB's. 

x-5=Uy+5}  (1) 

a?4-5s=r:3y— 15  (2) 

Subtract  (1)  from  (2),  and  we  have 

1 0 =%—  1 5—1—-         or ;  . .  V  =  1 K 

2     2 

(32.)  Let  a:=s=  the  number  of  bushels  of  wheat  flour. 

And      y=as  •<*  '^  barley    ^^ 

Then  the  cost  of  th«  whol«  will  be  expressed  by 

10a'-)-4y 
The  sale  at  11  shillings  v/ill  be  H^c-j-lly 

Now  by  the  coiiditi<ms, 

lOx^iij     :     llx^llif     :  ;     100     :     i43| 
Multiply  the  last  two  terms  by  4,  and 

iOx-^4:i/     :     n.r+!ly     :   :     400     ^     575 
t)ivide  the  two  last  terms  by  25,  and 

10a;+%     :     lla;4-lly     :  :     5'6     :     23 
^H^2y     :     llar-fllt/     :   :       S     ^     23  ^ 

ii5.r-f-46y=a:88a:+8% 
27a:=42y 
9;c=14y 
These  co-efficients,  9  and  14,  give  th«  lowest  proportion  la 
whole  numbers.     Th«  proportion  was  oaly  required. 

(33.)  Let  iOir-{ry  represent  the  number. 

Kow  the  question  gives  us         Q  =si^v 
And  Q'^^hy 

i(10^+y)=2;r-f  i         or .,y=3, 

INTEEPRETATIOJf  OF  NEGATIVE  VALUES. 

(Art.  55.) 

(4.)  Let  X  represent  the  years  to  elapse. 

Then         30+a:=3(154-.c) x^—1^. 

To  make  this  equation  true,  the  years  required  must  be  takea 
■mibtr<ic4ively^ 


ee  ROBINSOK'S  SEQUEL, 

(6.)  Let  x=  tlie  man's  daily  wages,  and  y=  the  son's, 
7a:-j-3i/=22  (1) 

6x-\-  y=18  (2) 

'2x-\-4y=40 

3x-{-  y=18     x=4,      y= — 2. 

(6.)  Let  j-ss  man's  wages,  y=  wife's,  and  z=  the  son's, 

10rc+  8y-f-  62=1030  cts,  (1) 

nx+lOy-^  42=1320  cts.  (2) 

16a:4-102/-f'122:=]385  cts,  (3) 

Subtract  (2)  from  (3),  and  we  have 

3a:+82=65  (4 

Multiply  (1)  by  5,  and  (2)  by  4,  and  take  their  difference, 
and  we  have  2x-\-\4z^=  — 130 

a-4-  72=     —65  (6) 

3a:-|-2l2=— 3-65 

(4) l'?Hr  ^^== ^ 

By  subtraction,  13z= — 4 '65 

2==_-4'5= — ^20, 
As  z  comes  out  with  a  minus  value,  it  shows  that  the  son  had 
no  wages,  but  the  reverse  of  it,  he  was  on  expense. 

(7.)  _  10ar-f-4y+32=1150  (1) 

9a;-f8y-f-62=1200  (2) 

7a:_f_6y-j-42=  900  (3) 

Double  (I),  and  subtract  (2),  and  11:?  =  1100 ar=100  ct*. 

(8.)  ^+1_3  X    _5 


V 

6      ♦ 

y+I 

7 

5a:+6=r3y 

(1) 

Ix 

=  5y+5 

(2) 

Add 

I2x 
3x 

=8y 
=2y 

(5) 

iract  (2) 

from 

(1)  and  we 
—2x+5=- 

have 

-iy—b 

(4) 

ar=— 10  by  adding  (3)  and  (4). 

y=— 15 
The  result  coming  out  minus,  shows  that  there  is  no  such 
arithmetical  fraction.     Algebraically,  however,  '_||  will  answer 
the  conditions. 


ALGEBRA.  69 

FINDING  AND  CORRECTING  ERRORS. 

This  subject  has  been  suggested  to  us  by  circumstances. 
Those  who  have  not  been  as  severely  disciplined  in  this  matter 
as  ourselves,  are  too  much  inclined  to  assume  that  the  error  must 
be  in  the  answer,  when  it  is  more  likely  to  be  in  some  portion  of 
the  data.  In  short,  we  believe  the  following  exposition  will  be 
of  use  to  many.     We  shall  illustrate  by  examples. 

1 .  A  young  man,  who  had  jiist  received  a  fortune,  sptnt  |-  of  it 
the  first  year,  and  |  of  the  remainder  the  next  year  ;  when  he  had 
$  1 420  left.      WJmt  was  his  fortune  ?  ^W5.  $  11 360. 

Let  x=-  the  income  ;  then  if  he  spent  |  of  it,  f  would  be  left, 
and  the  next  year  he  spent  |xi=ro- 

Then,  ——^-^=1420. 

•    8      10 

The  value  of  x  in  this  equation  is  ^38933}. 

This  result  shows  a  great  error,  somewhere.  If  an  error 
existed  in  the  answer,  it  is  probable  it  would  be  in  one  or  two 
figures,  at  most.  But  every  figure  of  our  result  differs  from  the 
given  answer ;  and  besides,  it  comes  out  with  a  fraction,  which  is 
against  probability. 

We  will  therefore  assume,  that  the  stated  answer  is  correct, 
and  that  the  error  is  in  1420, 

To  test  this,  write  11360  for  x,  and  a  for  1420  in  the  equation. 

Then,  a=('?— A^  11360=^- 1 136=852. 

\8     10/  4 

By  this  supposition,  1420  should  be  852  ;  but  we  can  conceive 
of  no  mistake,  either  in  the  writer  or  the  printer,  that  would  be 
likely  to  change  1420  to  852,  they  are  so  entirely  unlike  in  all 
respects.  We  therefore  assume,  that  1420,  as  well  as  the  answer, 
is  correct. 

Now,  it  only  remains  to  find  the  error  in  one  of  the  fractions, 
I  or  ^.     To  try  |  let  it  be  represented  by  m. 

Then,  (1 — m)=  the  portion  he  saved  the  first  year ;  |  of  this, 
or — - — =  the  portion  he  spent  the  second  year. 


TO  ROBINSON'S  SFQUEL. 

Hence,  ( \—m)x—^ilz:^}^=  1 420 

6(  1— «»)ar— 4(  1— m)a:=  1 420  •  5 
Or,  (1— m)ar=1420'5 

Write  the  value  of  a;  in  this  equation  and  we  have 
(!—»»)  11 360=  1420 '5 

1136. 
Whence,  »i=f 

Here  is  the  error ;  |  was  written  or  printed,  by  mistake,  in  the 
example,  when  it  should  have  been  f .  The  error  was  in  one 
figure,  only,  and  this  is  generally  the  case. 

2.  A  company  at  &  tavern,  wli&n  they  came  to  pay  thehr  hill,  found 
that  had  there  been  4  more  in  company,  they  would  have  had  a  shil- 
ling a  piece  less  to  pay  :  hut  had  there  been  8  less  in  company,  they) 
must  have  paid  a  shillihg  a  piece  less.  How  mawy  were  in  company ,, 
and  what  did  each  have  to  pay  ^ 

Ans.  24  persons.     Each  paid  7  shillings. 
Let  x=:  the  number  of  persons. 

y=  the  number  of  shillings  each  had  to  pay. 
Then,  .ry=  the  amount  of  the  bill. 
By  the  1st  condition  [x-\-A)[y — \)=Lxy  (1) 

By  the  ad         -         (;3r-.8)(y+l)=.ry  (2)^ 

Expanding,  and  omitting  xy  on  both  sides,  we  have 
43/ — X — 4=0 
— 8y4-.r— 8=0 
By  addition,  — 4y— 12=0  or,     y— — 3. 

This  value  of  y,  substituted  in  one  of  the  preceding  equations^ 
gives  3-= — 16. 

That  is,  there  were  16  less  than  no  persons  in  company,  and 
each  paid  3  less  than  no  shillings  ;  in  short  we  have  a  complete 
absurdity  in  all  respects.  No  change  in  numbers  expressed  in 
the  answer  will  remedy  the  matter,  and  indeed  with  the  present 
data,  no  other  values  can  be  assigned  to  a?  and  y. 

Now  to  find  where  the  error  is  in  the  data,  let  m  represent  4, 
and  n  stand  in  the  place  of  8,  in  equations  (1)  and  (2),  and  in 
place  of  X  write  24,  and  of  y  write  7. 


to  find  J 
X  and  y  | 

[  y=2,  or  1 


ALGEBRA.  Tl 

Then  (1)  becomes      (24+w)6=7-24  (3) 

And  (2)  becomes       (24— w)8=7-24  (4) 

Then  m=4.  and  »=3. 

Here  then  we  find  that  8  in  the  example  was  printed  in  the 
place  of  3.     This  correction  being  made  every  thing  corresponds. 

In  an  Algebra  recently  published,  I  find  the  following  example 
given  for  solution.     It  contains  an  error — find  that  en*or. 
3.  Given 

(  Ans.  ar=6,  or  3. 

'\-:i L_!l_= -        +rk     fin /I 

a;=y3  4-2. 

This  example  is  given  under  equations  of  the  second  degree  ; 
but  when  we  attempt  a  solution,  the  resulting  equation  will  pro- 
duce an  equation  of  a  higher  degree. 

As  the  answers  are  given  in  small  commensurate  numbers,  it 
is  probably  highly  probable  that  they  are  correct,  or  rather  should 
not  be  changed. 

Because  the  second  equation  verifies  with  both  answers,  we 
must  regard  that  as  correct. 

It  is  also  very  improbable  that  an  error  should  exist  in  regard 
to  the  radical  sign  of  square  root,  but  in  regard  to  the  exponent 
|,  an  error  there  is  very  possible.  On  this  supposition  we  will 
substitute  the  values  of  x  and  y,  in  the  first  equation,  and  write 
m  for  the  exponent ;  then  we  shall  have, 

2        ,  V6+2_      17 
(6+2)""^       ^  ^^-^^ 

Multiplying  by  ^/B  and  we  shall  have 

^+-?=1!  or,   lVi+i6=17 

8"  ^2      4  8"^ 

Whence,         878=8^=8"         That  is,        m=\ 

From  this  we  learn  that  the  printer  inverted  the  terms  of  the 

fractional  exponent.     This  being  corrected,  all  is  harmonious. 
We  shall  give  other  examples  in  finding  errors  as  we  naturally 

come  in  contact  with  them. 


7»  ROBINSON'S  SEQUEL. 

^  PURE  EQUATIONS. 

We  omit  all  the  examples  in  Robinson's  Algebra  to  the  17th, 
page  136.  From  thence  we  touch  all  that  can  require  any  notice 
in  a  work  like  this. 

(17.)  Divide  the  numerator  by  its  denominator,  in  each  mem- 
ber, and  we  have 

i--_^L_=i^ 1^ 


j6x-^2  4J6X+6 

Drop  1  and  change  signs,  and  clear  of  fractions,  and 

iej6x-{-24=Uj6x+30   .        Hence x=6. 

(18.)  Cube  both  members,  and 

Hence?      64-{-x^  — 8a;=  16+8a;+a;*         or a;=3. 


(19.)  By  clearing  of  fractions,         6-|-ar-|-^a;^-j-6a:=15 
By  reduction,         ^a;^+6a;=10 — x        Square,  <fec. 

(20.)  I z         I Z     3       J^ 


N  *+  V* —  ^/* —  V*=2 


\lx-\-  Jx 

Multiply  by    \x-{-  Jxy  and  we  have 

x-^Jx—Jx^—x=^Jx 

2X'\-2jx—2j~x^—x=3jx 

^^Jx=2jx^^ 

4x^ — 4xJx-\'X=^4x^ — 4x  af=x|. 

(21.)  Resolved  in  the  work. 

(22.)  Resolved  the  same  as  17. 

^__      2&      _^_ Ih 


[Ob= 
(  U.  136.  ) 


Hence,         6jaa;-\-10b=7  JaX'{'7b     or x= 


ALGEBRA.  73 

(23.)  Square  both  members  and  we  have 


Jx^-{-12=2+x             Square,  &c.  x=2. 

(24.)  Multiply  numerator  and  denominator  by  the  numerator, 
and  we  have  

(V5±l±V_4fZ=9. 
1 


Take  square  root  and  transpose  sj^x,  and^4a;-|-l=3 — J4x. 
Square,         4x-\-l=9—6j4x-\-4x.  Hence, . . .  .x=^. 

(25.)  Square  root  gives  a — x=Jb,  or  x=a — Jb. 
(26.)  Clearing  of  fractions  and  we  have 

4— 4a;2=3        x^  =  l-.         a;=±i. 

(27.)  Take  the  square  root  of  both  members,  and 

.2*1  . 

=-     or x=i5. 

x—1     2 

(28.)  Resolved  the  same  as  the  21st. 

(29.)  Clearing  of  fractions  we  have 

Jx'—9x-{-x—9=S6 

tjx^  — 9x = 4  5 — x 
a;2_9ar=452— 90a;+.^2 

8l2;=45-45 

9-9a;=5-9-5-9 a;=5-6=25. 

(30.)  Resolved  the  same  as  (17)  and  (22.) 
Dividing  numerators  by  denominators,  we  have 

3-^=3-i^- 
Va;-f2        V^+40  ^        ..         j. 

Drop  3  from  both  sides,  change  signs,  and  divide  by  5.  and 
clear  of  fractions,  and  then 

2^+80=21  J^-|-42.     Hence x=4. 

(31.)  Multiply  numerator  and  denominator  of  the  first  member 

Then,  _(>/HV^z::^=— 


14  ROBINSON'S  SEQUEL. 

Multiply  by  a  and  take  the  square  root,  and 

—     an 

Jx+Jx—a=     

Jx — a 

tjx' — ax-^-x — a=an 

Jx^ — ax=^[n-\-\)a — x 

Drop  x^ ,  and  divide  by  a,  and 

— x=^(n-\-\Ya — ^nx — 2a? 

l+2» 

(32.)  Resolved  in  the  work. 
(Art.  90.) 

(4.)  Observe  that  180=9-20       189=9-21.      Put  a=9. 
a:2y-|-a:y2_20a  •^ja.j.ya—.gla 

Multiply  the  first  equation  by  3,  and  add  it  to  the  second, 
and     a;3+3a;2y4-32:2/2_|_y3_8i«=a3 
cube  root,  a:-|-y=a=:9 

The  rest  of  the  operation  is  obvious. 

(6.)  Divide  the  first  equation  by  (x-\-y)  and 
x^—xy-\-y^=xy 
x^ — 2a;3/-|-y^  =:0                 or    x — y=^^. 
Hence a;=2     y=2. 

(6.)        x-\-y     :    a;     :  :     7     :     5  xy-\-y^^\tQ, 

5x-{-5y=7x 

5y=2x        or    ir=fy. 
Put  this  value  of  x  in  the  second  equation,  and 
^y^+y^  =  126 
7y2  =  126-2 
ya  =  18.2=36 y=d=6. 

(7.)  From  the  first  equation  we  have 

5x — 5y=s4y  5x=9y 


ALGEBRA.  7i 

181y2  =  l81  -25    or y»=25. 

(8.)  From  the  proportion  we  have 

5jj/=3jx       or ^5y=9x. 

The  rest  of  the  operation  is  obvious. 

(9.)  Extract  square  root  and 

ia:+i=3     or a;=7|. 

(10.)  From  the  first  proportion 

x-\-y=3x — 3y  or     4y=2x 

Hence 8y^=a;=*. 

8y3_y3_56  y=*=8  y=2. 

(11),  (12)  and  (13)  resolved  in  the  work. 

(14.)        f^::J^_6  or «-H=6 

From  the  second, .xy=5. 

(15.)  Divide  the  first  equation  by  (^-\-y),  and 
x^ —  xy-\-y^=2x7/ 
x^—<^yJ^y^=xyz=\Q  (1) 

Add  4xy  =64 

a;2+2a^+y2=         80=16-5 

Square  root,  x-\'y=4j5 

Square  root  of  (1)  a; — y=4 

2x       =4^5+4. 

(19.)  Double  the  2d  equation,  and  add  and  subtract  it  from 
the  1st,  then 

a;^  -\^2xy-\-y'  =a+2  J 

x' — ^y-\-y^  =a — 25 

x+y=:Ja-\-2b 


76  ROBINSON'S  SEQUEL. 

(Art.  92.) 

(6.)  Add  the  two  equations  and  extract  square  root,  and  we 

have  x^-\-y  ^  =  ±4  ( 1 ) 

Separate  the  first  member  of  the  first  equation  into  factors,  and 

we  have  x^(x^+]/^)  =  \2  (2) 

Divide  (2)  by  (1)  and  x^=z±zS  x=9, 

(6.)  Is  of  the  same  form  and  resolved  the  same  as  (5.) 

(7.)  Add  the  two  equations,  and  extract  square  root,  and  we 

3  3  

have  X*  -{-y  *  =  Ja-\-b 

3  /     JJ  3.  \ 

But  a;*,U*+yV=a 


rr=-     «'                or 

r     (     "' 

(u+6y 

\(a+b^. 

(8.)  Resolved  in  (Art.  90,)  of  this  Key. 

(9.)  Square  the  first  equation,  and 

x+2x^y^+y=^5 

(1) 

Difference,  ^.x^y""       =12  (2) 

Subtract  (2)  from  (1)  and 

By  evolution,  x^ — y^  =±1 

But,  x^+y^  =     5 

2ic2  =6  or  4. 


The  following  are  not  in  Rol)inson's  Algebra.    They  are  mostly 

from  Bland's  Problems.     We  shall  number  them  in  order. 

18 
(1.)  Given    x^-\-'3x — 7=a;+2+—  to  find  the  values  of  x. 


Ans.  +3,  —3,  —2. 


ALGEBRA.  77 

Reducing  x'-\-2x=9+— 

X 

Factoring  x'(x-{-2)=9(x-\-2) 

This  equation  will  be  verified  by  putting  x-\-2=0 

Then  will  a;=— 2 

Again  dividing  both  members  by  (ir-j-2)  and  x^=9 

Whence  x=zt:3. 

As  a  general  thing  we  shall  not  give  all  the  roots  to  equations. 
Imaginary  roots  we  shall  not  pretend  to  give,  except  in  rare 
cases,  or  unless  we  have  an  ulterior  object  in  view. 


(2.)  Given  ^^^-7^+^^-?-==^^  to  find  a;. 

^    '  axe 

Let  P=z^a-\-x         thenP^=a-f-^     ^^^  the   equation 

become*  — ~h — =-^^ 

ax         c 

Or  Px-\-Pa__Jx 

ax  c 

That  ia  P(a:+a)=^-ar^ 

c 

Whence  P^:=z-'x^ 

c 

By  taking  the  cube  root  we  shall  have 

By  squaring  and  replacing  the  value  of  P^  we  shall  have 

Let  the  known  co-efficient  (  ^  V  be  represented  by  m. 

Then  a-\-x^=mx  or,     a;= ^ 

m — 1. 

(3.)  Giren  (gjf  I^+(''+»)  •'=?  to  find  x. 


n  ROBINSOlS^S  SEQUEL. 

This  5s  the  same  as  example  2,  in  case  «=2,  therefore  we  may 
jump  to  the  conclusion  at  once. 

Thus,  a+x=^(^y^x. 

(4. )  Given    i«^'  +y ' = i^+y)^!/)  ^  find  the  values  of  x  and  y. 

Dividing  the  first  equation  by  («+y)  and  the  second  by  ay 
we  obtain 

«^— ^+2/^— a;y  (1) 

^+y=4  (2) 

Transposing  ccy  in  (1)  from  the  second  member  to  the  first 
gives  x^  — ^y-^y^  =0 

Whence       x — y^^O  or,         x^ssy 

These  values  substituted  in  (2)  give         ics=2        ys=2 
In  this  example  we  divided  one  of  the  equations  by  {x^y)^ 
therefore  (x-^-y)  must  contain  a  root  of  that  equation.  (See  theory 
of  equations.)     That  is,  ar-j-yz^O,     or,  xss: — y,  and  — y  substitu- 
ted for  X  in  either  equation  will  verify  it 

(5  )  Given    i^^'+^/M'^yC^+S^)  =^8)  to  find  the  values  of 
^    '^  ix^J^y^— Sx^—Sy^  =^12}  X  and  y. 

Ans.  xssi    or    2. 
y=2     or     4. 
Multiply  the  first  equation  by  3  and  to  the  product  add  thti 
second ;  then 

x^'^Sxy{x+y)'^y^tsti2l6  (1) 

Cube  root  a:+y=6  (2) 

By  squaring  and  transposing  (2)  becomes 

x^-^y^z=^36—2xy  (3) 

By  the  aid  of  (2)  and  (3)  we  perceive  that  the  first  equation 
is  equivalent  to 

36— 2a:y+62y=68         or     xy=Q     (4) 
From  (2)  and  (4)  we  find  x  aiid^. 


(6.)  Given 


ALGEBRA. 
1 


S      y        ~Q      X  7 


79 


To  find  the  values 
'  of  X  and  y. 


Put 
And 


shall  have 
And 


7(x—yy_7(x—yy^l 
4      y  4      X  9 

Ans.  a:s=|     y=i-. 

P^(x+y)^         F^^(x-^y) 

Q=(x—y)^         Q'''==(x—y) 
Divide  the  first  equation  by  f ,  and  the  second  by  I,  then  we 
P  ,  P^64 
~y       x^     63 

y       X      63 
Equation  (1)  reduced,  becomes 

(^+y)P^64 
xy  63 

F*_64 
xy      63 

xy     63 
Bivide  (3)  by  (4)  and  we  shall  have 

JL_=16 

Whence  F^2Q 

And  P3^8$3 

That  is  a;+y=8a: — 8y  or  9y='7x 

Ix^ 

From  this  last,  xy=s *  which  being  substituted  in  both  (3) 

y 


That  is 
Also  (2)  becomes 


(1) 

(8) 

(3) 
(*) 


ftnd  (4)  gives 
Whence 


9P' 


Ix^ 


.64 
'63 


or, 


9P'»^64 

2 


i>»==|,     Thatis(*+y)^=^    (6)- 


Substituting  ~  for  y  in  equation  (6)  and  we  have 
y 

3 


O) 

ROBINSON'S  SEQUEL. 

Cubing 

(•+i)"=i;- 

Tkatis 

••('+D'4:- 

Or, 

2232      83 
= — X 

92       93 

0 
4=:-x           or    a;= 

Ans 


The  following  solutions  refer  to  problems  in  Robinson's  Alge- 
bra, Chapter  V.,  Art.  93.     They  number  from  (5)  to  (13). 

QUESTIONS  PRODUCING  PURE  EQUATIONS. 
(5.)  Let  ic-|-2/=  the  greater  number. 

And       X — y=  the  less. 
Difference  2y=4  Sum  =2x 

2x{4xy)=1600       Hence ic=10. 

(6.)         Let  x-\-y=  the  greater  number. 
X — 1/=  the  less. 
2y     :    x—ij     :  :     4     :     3      or  ic=|y- 

(     a;2_y2)(  a;_y)=504 

(Vy^— y")(|y— y)=504 

(Vy')(iy)  =604    Hence  y=4. 

(7.)  Let  8a:=  the  length  of  the  field,  and  5x=  its  breadth. 

Then  = — =  the  acres. 

1^0       4 

Ix^  X  8a:=  the  whole  cost. 

^6x=  the  rods  around  the  field, 

13X26a:=  the  whole  cost. 

Hence  2a;3  =  13 -262;     or    a;=13.  8ar=104.     An9, 

(8.)  Let  5x=  the  length  of  the  stack. 

4x=  the  breadth. 

Then,  —=  the  height. 


ALGEBRA. 

5x*ix'-^'4x=  the  cost  in  cents. 
2 

Also, 

5x'Ax'9,^A=  the  cost  in  cents. 

Hence, 

5x'4x*ll'4x=5x'4.X'2M 

7x'2x=224         or 

81 


a;=4. 


(9.)  Put  a;*— 7=  the  number. 


Then,  x-^Jx^-\-9=9 


Jx''-{-9=9—x      or a?=4. 


(10.)  Let  X  represent  ^'s  eggs  ;  then  100 — ar=  ^'s  eggs. 

18  At        '  o      Tt,        . 

■■As  price.         _=^'s  price. 


100— re  X 

Hence,         _i^=?(100— ;r) 
100— a;    x^  ^ 

9a;2=4(100— a:)2 
3a;=2(100— <?;) ir=40. 

(11.)                Let  x-\-y=  the  greater  number, 
X — y=  the  less. 
2x=6     or     x=3 
(x-\-y)^=x^-\-Sx^y+3xy^+y^ 
(a:— y)  ^  =zx^ — Sx^y-}-3xy^  — y  » 
2x^             +6a:y2=72 
Divide  by  2a;  and  a;^+3y^  =  12.  Hence, y=l. 

(12.)        Let  x=  one  number. 

Then,      a'x=  the  other.  , 

a 

(13.)        Let  x^  and  y^  represent  the  numbers.  *♦ 

Then  a;^ +2/^  =  100  ^' 

«  +y  =  14. 


82  ROBINSON'S  SEQUEL. 

The  following  are  additional  problems,  and  for  the  sake  of  dis- 
tinction we  shall  mark  them  (a),  (6),  <fec. 

(a)  Two  men,  A  and  B,  lay  out  some  money  on  specvlatwn.  A 
disposes  of  his  bargain  for  $11,  and  gains  as  much  per  cent,  as 
B  lays  out ;  Ws  gain  is  $36,  and  it  appears  that  A  gains  four  timss 
a«  mtich  PER  CENT,  a*  B.     Required  the  capital  of  each  ? 

Ans.  A's  capital  $5,  B's  $120. 
Let      x=:B's  gain  per  cent. 

Then  4x=A*s  gain  per  cent ;  also  what  B  lays  out. 
Per  question,     100     :     ar     :   :     4a;     :     36 

4a:2=36-100         or     x=30. 
Whence,  4x     or     120  is  A's  gain  per  cent. 

Therefore,      220  :  100  :  :   11   :  ^'s  capital  =1^^= 5. 

^  22 

(b)  A  vintner  draws  a  certain  quantity  of  wine  out  of  a  full  cask 
which  holds  266  gallons  ;  and  then  filing  the  vessel  with  water,  draws 
off  the  same  quantity  of  liquid  as  before,  and  so  on  four  times,  when 
8 1  gallons  of  pure  win£  was  left.  How  much  wine  did  he  draw  each 
tim^?  Ans.  64,  48,  36,  and  21  gallons. 

Let  a=256.  x=  the  number  of  gallons  of  wine  drawn  the 
first  time  ;  then  a — a:=the  wine  left. 

It  is  obvious  that  the  wine  drawn  out  the  second  time  will  be 
found  by  the  following  proportion  : 

a     :     a — x     :  :     x     :     >       ^^  =  wine  in  2d  drawing. 

a 

Then,    (a — x) — ^ ^  =^ ^  =  wine  left  after  the  second 

a 
drawing. 


a 


Agam,      a     :    ^^ L    :  :    x    :     ^ — —^ — =  the  third 

a  a* 

drawing. 

Whence,  (^rf)'_(^=?I"*=(f=f)!=  the  wine  left  after 
a  a^        .         a^ 

the  third  drawing.     Whence  we  conclude  that '    ^^   ^      would   be 
the  wine  left  after  n  drawings. 

After  four  drawings,      ^ L  =81     by  conditions. 


ALGEBRA.  83 

(a— fl;)*=81-256-a2 
Square  root,  (a—  )2=9'16o=9- 16-266 

Square  root  again,    a — x=3  •  4  •  1 6 
That  is,  16-16— a;=12- 16 

Or,  16-16— 12- 16=4- 16=64=^. 

(c)  A  and  B  have  two  rectangular  tracts  of  land,  their  tenths 
heing  as  7. to  6,  and  the  difference  between  the  areas  is  150  acres  ; 
B's  being  the  greater.     Jifbw  had  A's  been  as  broad  as  B's,  it  would 
have  been  672  rods  long  ;  bid  had  B's  been  as  broad  as  A's,  'it  would 
have  been  900  rods  long.     How  many  acres  ivere  there  in  each  ? 
Ans.  A's  2100  acres,  Ws  2250  acres. 
Let         7a:=  the  length  of  ^'s  lot  in  rods, 
And         ?/=  the  breadth  of  the  same, 
Then,  lxy=^  tlie  square  rods  in  u4's  tract. 
Again,         let  Qx=^  the  length  of  i>'s  tract  in  rods, 
And,  v=  the  breadth  of  the  same. 

Then,       6ya^=  the  square  rods  in  ^*s  tract. 

By  the  given  conditions, 

6?;a;— 7a:y=150-160  (1) 

Now  had  ^'s  been  v  in  breadth,  it  Avould  have  been  672  rods 
long,  therefore 

Ql^v=lxy  (2) 

Also,  dOQy=Qvx  (3)  by  the  last 

given  condition. 

By  multiplying  (2)  and  (3),  omitting  common  factors, 
•    112-900=7^-2 

Or,  16-900=a;2 

Whence,  a-=4-30=120 

Substituting  900y  for  6y.c  in  (1)  and  120  for  x,  we  shall  hav€ 
900y— 840y=150-160 
Or,  60y=150-160 

y=400 

Lastly,  IlL2^:12?=2100  .I's  acres. 

160 


84  ROBINSOIf^S  SEQUEL. 

(d)  A  and  B  engaged  to  work  for  a  certain  number  of  days.  At 
the  end  of  the  time,  A,  who  had  been  absent  4  days,  received  $18.75, 
while  B,  who  had  been  absent  7  days,  received  only  $12.  Now,  had 
B  been  absent  4  and  A  7  days,  each  would  have  been  entitled  to  the 
sam£  sum.  ^ 

How  many  days  were  they  engaged,  and  at  what  rate  ? 
Ans.  They  were  engaged  for  19  days,  A  at  $1.25,  B  a^  $1  per  day. 
Let  ^=  the  time  or  number  of  days. 

a:=  the  daily  compensation  of  A. 
y=  the     "  **  B. 

Then  by  the  given  conditions 

(^— 4)a:=18a  (1) 

(/— 7)y=12  (2) 

{t--l)x={t-4)y  (3) 

From  (3)         ar='        ly.    This  value  put  in  (1)  gives 

{t:^y=\^  (4) 

t—1  ^  ^ 

Dividing  (4)  by  (2)  gives 

(^— 4)2^18|_  75  ^25 
{t—lf     !¥     12^     4^ 

Square  root  =_  or     t=\9 

t—1     4 

The  value  of  t  put  in  (1)  and  (2)  gives.  .ar=1.25,    y=I, 


SECTION   II. 

QUADRATIC  EQUATIONS. 

The  following  are  but  hints  to  the  solutions  of  Equations  in 
Robinson's  Algebra,  University  Edition.,  commencing  at  example 
10,  page  167. 

(10.)         Put  (ar— 4)2=y. 

Then,  ?=1+1? 

y        y^ 

y'— 8y-|-16=0         or. ..y— 4=0, 


ALGEBRA,  85 

(11.)        Multiply  by  16.     Rule  2. 

Then,     64x^-^16x^-{-l=39- 16+1=625 

8^e-|-l=25  a;«=3  a:=729. 

(12.)         Add  5  to  each  member. 

Then         (x^—Zx-\-5)-{-(x''—^x-\-5)^==16 

By  substitution,    y2_|_gy^9_25 .y=2  or  — 8. 

Hence    x^ — 2x-\-5=4r  x=i. 

(13.)         By  (Art.  99)  we  have  . 

361      19    ' 

-^=_-^  t=—6     t^=3Q 

19         !9 

361       19    ' 

-^— ^=db2.,. ar=152  or  76. 

19 

(14)     Observe  that  81^^  and  —  s^e  both  squares,  and  if  these 

x^ 

are  taken  for  the  first  and  last  terms  of  a  binomial  square,  the 
middle  term  must  be  9a;_.2=18. 

X 

This  indicates  to  add  one  to  each  member.      Then  extract  the 
square  root     9x-|--=±10.     Hence,   x=l  or — 1 

X 

(15.)     The  first  member  of  (15)  is  the  same  as  (14.) 
Hence,  add  unity  to  each  member    and  extract  square  root ; 

I     29 

we  then  have  9ar-4--=— +4 

X        X 

9x^-^ix=2Q  Put  x=-. 

9 

«2_4^^28- 9=252 

«— 2=  ±16 x=<i. 


86  ROBINSON'S  SEQUEL. 

(Art.  105) 

(4.)         Multiply  every  term  by  x,  and    . 

ar4_|.8a;3-|_i9;r2_i2a;=o. 
Operate  for  square  root  thus  ; 


Divide  by  {x^ — ^x),.  aud 

ar2_4^4-3=0 a;=l  or  3. 

But  the  factor  a;^ — 4ar  gives  a:=0  or  4. 

(5.)         a;^— 10a:3+35a:^— 60a;+24=0  (a;^— S^r 


2ar2_5ir)     — 10a:3-|-35a:2 
— 10a:3-f-35a;2 


10ar2_50a;+24 
(ar2_5x)2+10{a;2— 5ar)+24=a 
If  we  add  unity  to  each  member,  we  shall  have   complete 
squares.     Extract  the  square  root,  and 

x^ — 5a*= — 4  or  — 6. 
From  these  two  equations  we  find  a;=l,  2,  3,  or  4. 

(6.)     By  mere  inspection  we  perceive  that  this  equation  can 
take  the  form     (f —xf —{x'' —x)^\?>1. 

y^—y=\o'2.  y=12or— 11. 

x^—x=\^  or— 11. 
If  we  take  — 11,  the  value  of  x  will  become  imaginary.     12 
gives  a:=4  or  — 3. 

(7.)     This  equation  may  be  put  into  this  form  : 
(if—cy^  )-^(^f —cy)=c'^ 
from  which  the  reduction  is  easy. 


ALGEBRA.  07- 

(Art.  107.) 

(3.)     Taken  from  the  work  we  have 

{a-|-l  )x^ — a'^x=a^ 
Or,  (a-\-l)x^={x+l)aK 

Both  members  are  of  exactly  the  same  form,  and  of  course 
the  equation  could  not  be  verified   unless  xz=:a. 

EXAMPLES. 

(1.)    x^'+Ux^SO.     Multiply  by  4,  &c. 
4a;2-(-^+l  12  =329+121=441 
2x+U=  ±21 x=5  or  —16. 

(2.)     Drop  2a;  from  each   member,    and  divide  by  3  ;  then 

x—l     x—2 

X — = 

x—3     ^  2 

Clearing  of  fractions  and 

2x^  —Sx—2x-{-2=x^  —3x—2x-{-6 

ir2_3a:=4.                 Put     2a=3. 
Hence,  (Art.  106) x=4  or  —1. 

(3.)     Multiply  the  equation  by  6x  ;  then 

fi'y2 

J^^-L.Gx-\-6=13x 
x+\^    ^ 

x+1^ 

6x''+6x-{-6=7x''+7x 
Hence  x^-\-x=e ' x=2  or  —3. 

(4.)     Clearing  of  fractions 

70a;— 21a;2+72a;=500— 150a; 

21a;2— 292a;=— 500. 

Or,     21a;2— 42a;=250a;— 600.    or  21a;(a;— 2)=250(a;— 2). 

(6.)     Put     ('?+y)=a;.     Then 
\y       / 

a;2-|-a;=30.         Or,         a;=5  or — 6. 
Now,  (5+y^=5or— 6 


fr.  ROBINSON'S   SEQUEL. 

y»— 6y=— 6,     or  y^-\-ey= — 6 

2y— 6=  del y=3  or  «. 

(6.)      Put  a:^=y  ;     Then  y^-\-'7y=44 

4y3_|_^^49=226 

2y+7=±15                        y-=4or-.ll. 
a;=(4)^  or  (—11)* 
(7)    a;2+a;=42.     Hence  a;= 6  or —7. 
That  is  y* +11=36  or  49 y=5  or  ^38. 

(8.)  11— f+!^=^ 

^    ^  ic— 7     3 

33a;— 23 1— 3a:— 21 =a;*— 7a; 
a;2— 37a;=— 252 

4x'^A+3V  =1369—1008=361 
2ar— 37=  ±19 a?=28  or  9. 

(9.)  3a;2— 9a:=84 

12 


36a;2— .^+81  =  12-84+81  =  1089 
6a;— 9=  ±33. 

(10.)     Clearing  of  fractions  we  have 

2a:+27a;=16— <i: 

3a?+2Va;=16 
Multiply  by  12,  &c. 

6Va;+2=±14 .a;=4  or  7f 


(11.)  ?(^=li)+4^=26 

?(?^1I)+2.=  13 
ar— 3      ^ 


6x— 33+2a;*— 6a;=13a;— 39 
2a:2— 13a:=— 6 

16a;2— ^+132=169— 48=121 
4a;— 13=  ±11 


ALGEBRA.  -  89 


(12.)     Multiply  by  x^  and  we  have 


9 


1  1^2 


9 
lla;2— 54a?=— 63 

Put  a;=— ;  then  w2__54^^_693 

«2_54^_|.272_36 

u — 27==b6 w=33  or  21. 

(13.)     Clearing  of  fractions  we  have 

_a;2=27a:— 28 

x''+27x=2Q.  Put2a=27.  Jj^-. 

<c2-f-2aa;=2a+l  '     "  ™ 

a;+a=±(a4-l) «=1  or  —28. 

(14.)     Given  mx^ — 2mxjn=nx^ — mn,  to  find  x. 
By  transposition,  ma:^ — 2mx  J n-\-mn=nx^ 
Square  root,  Jmx — Jmnz=.±:.Jnx 

By  transposition,         {Jm^Jn)x—Jmn 


The  following  are  not  in  Robinson's  Algebra.  They  are  too 
severe  for  learners  in  general,  and  are,  therefore,  not  proper  in  an 
.elementary  work. 

We  commence  again  with  No.  1. 

(1.)     Given    a;^-|-         =~2^-|-a;°,  to  find  the  values  of  x. 
We  observe  that  the  lowest  root  of  x  is  the  6th  ;  therefore, 

i.  JL  2  ^  7  i 

put   a;*=y;  then  x^=^y^.     x=y^ .  x^=^y^ .  x^=7/^.  x^=y^*. 


90.  ROBINSON'S  SEQUEL. 

Then  y^^-X-^z=?Lu.ys 

Whence,  yi»=66+y9. 

Therefore,  y^— i=±V-     Or,  y^  =8  or  —7. 

y3=2  or  (—7)3. 

2 

ar=y^=4,  or  ( — 7)^.     Ans. 

2 

(2.)     Given    *  /_L-4-^/i=5z!^l>  to  find  the  values  of  x. 

^  X  X  X 

Ans.  x=  1  or  — V 
Multiply  by  x,  and  then  we  shall  have 

'(•Vi)-H(■^/i)— •• 

Placing  the  value  of  x  under  the  radical  signs,  then 

.  ya;2+3/a;2=3— A 
That  is  a;^+a:^=3— ir^ 


2a;3+a;3=3. 
Whence,  a;3=il  or — f. 

Whence,  a;3_ior— |.     Or,  ar=:l  or— ¥•  Ans. 

(4.)     Given  ^ar— 1)'+ A— Ay=^  to  find  a;. 

Put  F=(x—-y  and  ^  =  A— i)  '  ;  then 

P+Q=x  (1) 

Multiply  this  last  equation  by  (P — Q),  then 

ButP2_g2— (^_1).  therefore,   a; (P— §)=(«— 1) 
Or,  P-^=l-i  (2) 

X 

Add  ( 1 )  and  (2),  then         2P=  (^— ^) +^ 

That  is,  2P=P2+1 

Or,  P2_2P+1=0,  or  P— 1=0 

Whence,     a?— 1=1,  or  x=^{l±:j5). 


ALGEBRA. 

(5.)     Given     (x^- 

->)*+  (••-: 

values  of  x. 

91 


We  observe  that  this  equation  is  in  the  same  form  as  the  pre- 
ceding, and  would  be  identical  if  we  changed  x^  to  x,  aMo  1. 
Therefore  the  value  of  ar^  in  this  equation  will  be  of  the  same 
form  as  the  value  of  x  in  the  last  example,  except  it  will  contain 
the  factor  a^,  because  the  square  root  has  been  once  extracted : 

That  is  x^=—{lzh^5), 


=-(^)' 


But  this  conclusion  is  too  summary  to  satisfy  the  young  algebraist ; 
therefore  it  is  proper  to  take  some  of  the  intermediate  steps. 

then  the  equation  becomes 

P-\-Q=~^  (2) 

xMultiply  (2)  by  \P — Q),  then  we  have 

a 
But  the  value  of  {P^—Q^)  drawn  from  (1),  is  (a:^— a^); 

therefore  ^  (P—  ^)  =  x^—a^ 

a 

or  i>_§=a-?l  (3) 

X'' 

By  adding  equations  (2)  and  (3),  we  find 

2i>=«-^+?!  .  (4)        ■ 

x^      a 

Multiply  this  equation  by  a,  then 

2aP=  a''——+x^ 
x^ 

that  is,  2aP=a^+P^ 

or  0=a2— 2aP+P2. 


92  ROBINSON'S  SEQUEL. 

Square  root,  0=  a — P,     or  P  =  a 

From  the  first  of  equations  (1),  we  find 


a     « 
.x^ — — 


,2 


From  this  equation,  we  find    x=^-=^a(.      ^  -\  . 

(6.)     Given     x^{\+l-y^{^x^-\-x)=10 ,  to  find  the  val- 
ues  of  X. 

Observe  that  (^x^+x)  =  ^x^{\-\-—).    Put  (14-J_)=y  ; 

^x  3a; 

then  the  given  equation  becomes    x^y^ — 3a;2y=70. 

—       9     289 
Completing  the  square,  x^y^ — 3a;^y-|--= 


x^y ^=db — 

"^      2  2 


ic2y=10,  or — 7 

That  is  a;2  4-^=10,  or —7, 

^3 


Whence        x=d,  or  —  V»     or  1(1=^^—251). 

/7^     Given         5^-^Jx_^^{x-2jx)  W-2,x+4. 

^   *^  ^+27^  6+V^  (^+2V^)  (6+V:r) 

to  find  the  value*  of  x. 

Multiply  by    (6+ ^a:),  then 

x-\-2jx  ^  ^    ^^    x+SLjx 

Multiply  by  (x-{-2jx),  and  we  shall  have 

9(S6—x)=23(x^—4x)-\-7x^—3x+4 
Reducing,  15x^—4Sx=160.    Whence  a;= 5,  or — ff. 


(8.)     Given       x^ -^^^4-15=^^^1^^,    to  find  the  values 
^    ^  2^  16       x^ 


of 


ALGEBRA.  93 

By  transposition,    a;^-|-15-f- — = + — ■ 

Add  1  to  each  member,    (see  Robinson's  Algebra,  Art.   99,) 

then  .>+16+^=?^^-+^+l 

^     ^a;2        16     '2^ 

By  evolution,  x+-  =  ±  (~+l\ 

X  \  4        / 

O  n» 

Taking  the  plus  sign,      _■'  =  — 1-1 .  ( 1 ) 

Taking  the  minus  sign,  -  =  — — — 1.  (2) 

X  4 

From  (1)  a;*+4ar=32.     Whence  a;=4  or  —8. 

From  (2)  92;2-)-4ic  =  — 32,  and  x^^'Z^^y.IzI} 

y 

(  9 . )     Given     {x^^sy  — 4a;2  =  1 60,  to  find  the  values  of  x. 
Subtract  20  from  both  sides,  then 

'       (a;24-5)2_4(a;24-5)=140. 

Whence,  x^  +5—2=:  ±12. 

Therefore,  x=  ±3  or  ±<y — 15. 

( 10. )     Given      — + - =^—1= ,  to  find  the  values 

^  ^  (a;2_4j2^(^2_4)        25^2 

of  x, 

rf-2 

Multiply  by  a;  2,  and  put    _— r-==y  :     then 

2  \  a       351 

Whence    y+3=  db V  •        y=  f  or  —V  • 

mt,  X  •        ic^         9  x""  39 

That  IS =-,  or = — — 

a;2__4     5  a.2_4  5 

x=:  ±3,     or  a;=  ±VH- 

(11.)     Given     (ar— 2)2— e^a:  (a;— 2)  =  24— Har+lS^a:,    to 
find  the  values  of  x. 

Expanding  and  reducing,  gives 

a:2— 6a;^a;=20— 10ar+3V^. 


94  ROBINSON'S  SEQUEL. 

Add  9x  to  both  sides,  and  the  first  member  will  be  a  square  ; 
that  is, 

X  2  —6x  Jx-\-9x = 20— <i:+3  Jx. 
Or,  (S^a;— a;)2=20— a;+37a; 

'     Now  put         3jx — x^=y  ;  then 

y2=20+y. 
Whence,  y^=5  or  — 4. 

Then  x—3^x^^,   .  or  —5. 

Whence,   :c=16,  or  1,  or  ^.^tzllz:! . 

2 

(12.)     Given     (4x+iy-]-4x^(4x+l)=19\2-^(l0x-{-3x^)io 

find  the  values  of  x. 

Add  4x  to  both  sides  ;   then 

(4x-\-iy-\-4x^(4x+l)+4x^l9n—6x—3x^ 
That  is  [  {4x+\)-{-2^x]-^-{-Gx+Sjx=1912 

Or,  [  2{2x+Jx)+l  Y+3{2x+Jx)  =  1912. 

Now  put  y=2x-\-Jx  ;     then 

(2y+l)  =  +3y=1912 
Or,  4y2+7y=:1911 

64^2+^-1-49=1911  •  16+49=30626 

8y+7=±175  * 

y=21,    or  — -V- 
That  is,  2^+7^=21,    or  —  V- 


From  these  last,  we  find  ^-=9,  or  V,  or      ^^"^J^JJl 


3;r 


(13.)     Given     8a;2^13=  -+  »j6x^-\-52x^ ,  to  find  the  values 

of  X. 

Double  the  equation  and  remove  the  factor  x^  from  under  the 
radical  sign ;   then 

16a:2— 26=3^+20:^6^^+62', 


That   is,  16.c2  =  (3.r+26)  +  2;r72(3x+26). 

Now  put  9/=  J (3x-\-26)  ;     then  the  equation  becomes 
iex^^ij^+(2j2)xy 


ALGEBRA.  95 

Add  2a;^  to  both  members,  and 

1 8a;2 = 2/2  _|_  2  ^2  •  iry+2a;«  • 
By  evolution,         Sxj2=y^j2  (a) 

Or.  2xj2=y=^3x+26 

By  squaring,  8x^=3x-\-26. 

This  equation  gives    a;  =  2,     or  —  y . 
By  taking   the  minus   sign   to  the  second  member   of  (a), 

we  would  find  x-= ^ . 

64 

(14.)     Given     4x''-\^21x+Sx^  J7x^—5x=207'^^f-,  to  find 

o 

the  values  of  x 

4x^ 
Transposing    — and  removing  the  factor  x  from  under  the 

radical  sign,  will  give 

I6x^  

-y-+21a;+8^77;c— 5=207 

Subtract  16  from  each  member,    then 
16x^ 


3 


■(21a:— 16)+8a;  77a;— 5=192 


That   is  l^+3(7a:—5)4-8a:77a:— 6=64-3 

o 

Put  y=:J{lx—5). 

Then  15^+32/2 +8ary=64  •  3. 

*  3 

Clearing  of  fractions,  and  changing  terms, 
16a;2+24a:y+92/2=64-9 
By  evolution,         4a;+3y  =  8  •  3  =  ±24 
That  is  4a:+37(7a— 5)  =  ±24 

Taking  the  plus  sign  and  transposing  4ar,  we  have 
37(7a:— 6)=(6— a;)4 

By  squaring  9(7a:— 5)  =  (36— 12.r+a;2)16 

Beduced,  16a:2_-265a:=— 621 . 


96  ROBINSON'S  SEQUEL. 

This   equation   gives  x=3,  or    — V/.     By  taking  — 24,  we 
obtain  ^^'^3±3V(-2567) 
32 

(15.)     Given  a^b^z^—4{ahyx^^=(a—^yxi',  to  find  the 
values  of  x. 

Put  P^^ajr  (1) 

And  Q^=xi^  (2) 

mfn 

Then        P^Q^=x^  (3) 

And  P^=a;2-«»  (4) 

Substitute  these  quantities  in  the  given  equation,  and 

aH^F''—4{ab)^FQ={a--by  Q^        (6) 
Kow  let  P=iQ 

Then         a'^bU^  Q'''-^4{ab)^iQ^={ar-by  Q' 
Dividing  by  Q'^  gives 

a^Pt^—4(ab)H=(a—by=a''—2ab+b^ 
Add  (4ab)  to  the  first  member  to  complete  the  square,  (see 
Art.  99,  Robinson's  Algebra.) 

Then         a''b^t''—4(ab)H-\-4ab=a^+2ab+b'' 
By  evolution,  abt — 2^(a5)=a-|-5,  or  — a — b 

Whence  abt=:(a+2jab+b)=z(Ja^Jby 


Or,  f-U<^+J^r   or,n(y-^irN^ 

ab  '  ab 


^    .            ,     P      xTi  "-" 

But  ''= — = =  icamn 

Q      -i- 

*      ar-2 » 


Therefore,     .•^■=(Vl±^^  or,  =1>=V*): 
ab  ab 


Whence, 


JM^|s„  j-ws^js 


ALGEBRA.  97 


SECTION   III. 


QUADRATIC  EQUATIONS  CONTAINING  MORE   THAN  ONE 
UNKNOWN  QUANTITY. 

We  commence  by  showing  the  outlines  of  the  solution  of  the 
(3),  (4),  (6),  (6),  (7),  and  (8)  equations  in  Robinson's  Alge- 
bra, Art.  (Ill),  page  182. 

(3.)     Futx^=F,  and  y^=Q.     Then  the  equations  become 
F+Q=Q  (1) 

Square  (1)  and  we  have   P^-\-2FQ-\-Q''=64^  (3) 

Subtract  (3)  from  (2),  and  we  have  P^  Q''—2FQ=195, 
Hence,  P$=15or— 13. 

Now  we  have  P-\-Q=8,  and  PQ=15,    whence  P=5  or  3, 

and  ^=3  or  5.      That  is,  x^=5  or  3,  &c. 

(4.)     Puta;^=P,  and  y^  =  Q;    then   the   equations   become 
P^  +  Q''+P+Q=26,    andP$=8 

2PQ =16 

(>+^)2+(P+$)=42.     Hence,    P+Q=6. 

(5.)     Put   — =:u:     then   u^-\-4:U= —     w==_  or — — . 
The  remaining  operation  is  obvious. 

(6.)  Given  y^ — 8a;^y=64,  and   y — 2a; ^y^  =4,  to  find  a:  and  y. 

To  both  members  of  the  first  equation  add  16x,  and  to  the 
second  add  x,  to  complete  the  squares  ;  then  extract  square  root, 
and  we  have 

y— 4a;2=4(a?+4)^  and  y^—x^=  (a;+4)^ 
Four  times  the  last  equation  subtracted  from  the  preceding, 
gives  y — 4y2=0.     Or, y=16. 


•8 


ROBINSON'S  SEQUEL. 


(7.)     Multiply  in  the  first  equation  as  indicated,  and  subtract 
the  second  equation  ;  we  then  have 

«+y+2^'y'=25     or  x^+y^=:6 
But  from  the  second  equation  we  have 

(a;2_(_y2)a;2y2=3o.     Hence, x^y^  =  6 

3.  2^  X  JL  9 

(8.)  Divide  the  first  equation  by  y^,  and  x^=2y^,  or  y^=^x'^ 
This  put  in  the  second  equation  gives 

X  z  f 

a;T_16a;3+64=64— 28=36. 


We  continue  this   section  by  adding  other  and  more  severe 
equations,  commencing  with  number  one. 

(1.)     Given      -I  oI'^'^^^^^Ta  I    \     to  find  the  values  of 
^    ^  (  28 — y  =  x-\'^»Jx  J 

X  and  y. 

By  adding  the  two  equations,  omitting   16  on  both  sides,  gives 

Squaring,  144 — Mjy-{-y—\Qx  (1) 

Multiply  the  first  equation  by  16,  and  substitute  the  value  of 
— 16a;  from  (1),  then  we  shall  have 

16y— 16^^=266— 144+247y—y 
Whence,  17y— 40^^=112 

>/y=f^±V(H|-^+lff)  =  n±H  =  4or-^. 
Therefore,  y  =  1 6  or  |f  ^ . 

These  values  of  y  put  in  the  first  equation,  give 
x=Jy=A,  or  my. 


(2.)     Given 


y  J(^x-\-y)  17         1  to  find  the 

(x+yy'^       y      ^^^Jix+y)    [values    of 

x=y^^2  J  X    and   y. 


ALGEBRA.  99 

Mnltiply  the  first  equation  by  >/(^+y)»  *^^^ 
y    _|_^+y^^7 

^+y     y      4 

Clearing  of  fractions,  and 

Reducing,  4a;2  ^^^xy-^-^y^ 

Adding,  to  both  sides,  (Robinson's  Algebra,  Art.  99)  and 

4  4 

By  evolution,  :?^=±(— +3^/) 

Whence        a;=3y   or  — fy. 

These  values  of  a;  put  in  the  second  equation,  readily  give 

«=6,  or  3,  or  9T3V(-n9) 
32 

,     y=8,orl,or-^-^V(-"9_) 
8 

(3.)     Given       j  a:+4^^+4y=21+8Vy+4V(a:y)  \    ^  ^^ 
and       t  Jx-^Jy=Q  ) 

the  values  of  x  and  y. 

From  the  first,  x--'^J(xy)-\-^y—2\-\^^Jy—Ajx 

That  is  {^Jy—JxY  =21+4(2^y— 7«) 

Let  P=^Jy — ^a: ;     then 

.^  P^— 4P=21 

^^^■L  -P=2  ±  726=7  or  —3. 

tR^        Zjy—Jx=l  or  —3.         But   Jx^Q—Jy, 

Therefore  Sjy-^=7  or  —3. 

7y=  V  or  1. 

y  =  -If  ^  or  1. 


(4.)     Given       i  Sx+ljxy^+9x^y==.(x^x)y  1 

and       (  6x+y  :  y  :  :  x+5  :  3  J-    to  fand  the 

values  of  x  and  y. 

From  the  first         9a;rf-2  Jny^-^9x^y=:3xy—y 


100  ROBINSON'S   SEQUEL. 

That  is         (y^9x)+2jxy  {7/+9x)^' =^3xy 
Add  xy  to  both  members  and  extract  square  root,  then 
Ji/+9x+J^=2jxy  (1) 

Whence  y-\-9x=xy  (2) 

From  the  second    — 2y-j-18a;=ary  (3) 

By  subtraction,  3y — 9a; =0 

Or,  y=3x 

This  value  of  y  put  in  (2)  gives  12^=3a;^. 
Or,     x=4.     Whence  y=12. 

By  taking  the  minus  sign  to  the  second  member  of  (1),  other 
values  of  x  and  y  can  be  found. 


(6.)     Given       \  x-^-y^-J  "7"^— i     to  find  the   values 

and      Ix^+rJiT'-'    ^~^)     °f^aidy- 
The  first  cleared  of  fractions  is 

x^ — y^ — Jx^ — y^=iQ 

Whence,  -      Jx^—y^=3,  or  —2 

a?=±5,    or    =1=371 


(6.)     Given       (  J^x+xYJ^y-+J{\-xY+y-=4  )         ^ 
and       (  (4_;p2-)2^j8_4y3  f 

the  values  of  x  and  y. 
From  the  first 


Squaring, 

\^2x-{-x^-\-y''=\Q—^J(\—xY-\^^+\—^-\-x^-\-y'' 

Reducing,  a;= 4— 2  /(I— arj^+y^ 

Transposing  4  and  squaring,  gives 

a;2_8:y_|-lC=4(  1— 2a:+a;2 +y2  ) 

Reducing,  12=3a;2+4y2  (l) 

That  is       4— a;2=^y!.,    U—x^Y=:]^ 

Comparing  this  last  result  with  the  second  equation,  we  per- 
ceive that 


ALGEBRA.  101 

l^'.+4y'=16  (2) 

Add  I  to  both  members,  (Art.  99,  Algebra,)   then 

9     ^  ^  ^4      4 

T,  1  .•  4y2     3         9 

By  evolution,  _^_-|-_==±- 

3*22 

Whence  4y2=9,    or  — 18 

y=dbf,    or   ±fV— 2 

The  value  of  4y^,  that  is  9,  put  in  (1),  gives  x=\. 


.2 .,2 


(7.)     Given       {^+^'^I^=^l-t-r^- 

^  x—Jx^'—y^      4     x^Jx^'—y^  >       to   find 
and       (  ^2  ^xy=52—Jx^+xy-\-4  ) 

the  values  of  x  and  y. 

Add  4   to  both  members  of  the  last  equation,  and  transpose 
the  radical,  then 

(x^-^xy+4)+(x^+xy+4)^=56 
This  is  a  quadratic,  and 


Jx^+xy+4+^  =  ±V^F  =  ±V 
Whence,  Jx"-\-xy-\-4=7,  or  — 8 

a;2-|-ary=45,  or  60.  (1) 

Now  take  the  first  equation,  and  multiply  numerator  and 
denominator  of  each  of  the  literal  fractions  by  its  numerator, 
then 


Expanding  and  uniting,  and  we  have 

4 

16a;2=25y2 


4a; 

Ax  —  ±:  5y,  or  y=  ± — 

5 


10«  ROBINSON'S  SEQUEL. 

This  value  of  y  put  in  (1),  gives 

ar2-|-!^'_=45,  or  60.  (2) 

43*2 

Also,  a;2— Zf„=45,  or  60,  (3) 

6 

From  (2),  9a;2=9*5-5.     Or,     a;=  ±5. 

Or,  9a-2=25-12=25-4-3 

3ar=5-273.     Or,     a:=dbl0^i 
From  (3),  a;2=9-6-5.     Or,     a;==  dbl5 

Or,  ir2=300.     Or,     x=  ±10^3. 

Here  we  have  8  different  values  of  x,  each  of  which  being 

4a; 
substituted  in    y=rfc — ,  will  give  8  different  values  to  y. 
5 


(8.)     Given      f    h+V^  \    -^       -^^   /_4^  1  to  find 
and      I  V_^<5^=^=y+i  [andy. 


Clearing  the  first  of  fractions,  gives 

x-\-y^+^Jx=:^xy^  (1) 

In  the  second  equation,  multiply  the  numerator  and  denomina- 
tor of  the  fraction  by  the  numerator ;  then 

Multiply  by  (y+1),  then  extract  square  root,  and  we  shall 

have 

Jx+Jx—y—\^y+\  (2) 

Or,  Jx—y—\={y+\)-^Jx 

By  squaring,         x—y—\=y^-\-9.y-\'\ — '^Jx{y-\-\)-\'X 
B^duced,  (i=^(y''+y)+^y-\'^—^Jx{y-\-\) 

Dividing  by  (y+1),     0=y+2— 2V^  (3) 

As  we  can  divide  by  the  binomial  (y+1)  without  a  remainder, 

it  follows,  by  the  theory  of  equations  that  (y+1)  contains  a  root, 

that  is  y-|- 1=0.     y= — 1. 

Corresponding  with  this  value  of  y,  equation  (3)   or  (2)  will 

give  the  value  of  :r.     2jx=l,     ar  =  ^. 


ALGEBRA.  108 

To  find  other  values,  we  must  continue  the  solutions. 
Return  to   equation  (1)  and  extract  the  square  root  of  both 
members,  and  we  shall  have 

Jx-{-y=  ztyjx  (4) 

From  (3),  2jx=y+2  (6) 

Double  (4),  and      2jx-\-2y=  :±S.Jx{y)  (6) 

That  is,  y+2+23/ =y2  +2y 

Or,  y^ — y=2'     Whence, y=2,  or  — 1. 

The  value  — 1  we  found  before  ;  which  shows  two  roots  equal 
to  — 1.  The  other  value  2,  put  in  (6),  gives  a;=4.  If  we 
take  the  Djinus  sign  in  (6),  we  shall  have 

y+2+2y=— 2/2— 2y 
Or,  2/2+5y=— 2 

Whence,  y=  — f  =h^  ^17 


(9.)     Given 
and 


2a;2 X    _1  I  ues  of  a;  and  y. 

The  first  equation  can  be  put  in  this  form 

The  solution  of  this  quadratic  gives 


or 


Whence,  ^=16,  or  ??-.    y=16a;.     Jy=^Jx. 

X  \Q 

Substituting  the  values  of  y  and  Jy,  in  the  second  equation, 

we  find 

x__^     X     J_l 

8     127^     3 


iH  ROBINSON'S  SEQUEL. 

The  double  is  ^-Jjx=- 

Add  3^  to  both  members  to  complete  the  square,  (Art.  99, 
Algebra,)  then 

By  evolution,  ^  Jx — ^=±f 

Whence,      x=4,  or  V  >  but  y=16a;=64,  or  ^S■. 
If  in  the,  second  equation  we  write  ^j^x  for  the  value  of  y, 
and  V  J^  for  the  value  of  Jy,  we  shall  find 


•*' 64    >     ^^      144' 


values  of  x  and  y. 

Transposing  2a:^y^  in  the  the  first  equation,  and  we  have 

By  evolution,  x^ — ^y^  =  ±(l-|-^y)  (1) 

The  second  equation  can  be  put  in  this  form, 

(23,='+l  )(*+!)  (2) 

Taking  the  plus  sign  in  (1),  we  can  put  it    into    this  form 

x^—xy+y^=2y^+l  (3) 

By  the  help  of  (3)  we  perceive  the  equal  factors  in  (2).     Sup- 
press them,  and  (2)  becomes 

x-^y=x-\-l.     Or  y=l. 
This  value  of  y  put  in  (1),  gives  a?=2,  or  — 1. 


(11.)     Given 
and 


£J^-40y^=136-y»J^'-!^  I    u>    find 
^  "3'      Itheval- 

yyy         '    y^    y  J   andy. 

It  is  obvious  that  the  first  equation  can  be  put  into  this  form 

a;2y2_8()y  2  =272— 2^^0:^—272 
By  transposition 


ALGEBRA. 


106 


(a;2y2__272)+2y7a:2y2_272=80y2 

By  adding  y^  to  both  members,  and  extracting  square  root  we 
have 

(a;2y  2_272)  2+y=  ±9y 
Whence,         x^'i/'^—272=e4y\  or  lOOy^  (i) 

Clearing  the  second  of  the   given  equations  of  fractions,  and 
reducing,  we  have 

x^y'^—SBxy^Se  (2) 

Put  2a=35,  (see  Art.  106,  Robinson's  Algebra.) 
Then  x^y^ — 2cui;y=2a-\-l 

Adding  a^,  and  taking  square  root,  gives 

xy — a=ztz(a-\-l) 
Whence,  xy=z(2a-\-l)=36,  or —1  (3) 

These  values  of  xy  put  in  (1),  give 

64y2  =36 -36—272,   or  6V  =  _271 
64y2  =  i024,  or  8y=±32.     y=4,  or  —-4. 
These  values  of  y  put  in  (3),  give  x=9,  or  — 9. 
Again,  by  observing  (1),  we  perceive  that  we  may  put 
100y2  =  i024,  or   10y=  ±32.     2/=3.2,  or  — 3.2. 


(12.)     Given 
and 


2y^—^Jx 


+  2jy-—\Jx^ 


^Jx 


I  V^+V8(2/->/^)— 4=y+l 


to  find 
the  val- 
ues of« 
and  y. 


Put    7^2. 
Then 


[^x=P,  in  the  first  equation. 

^^     I   2P  —  ^v^ 
Jx^  2 


4F^-]-iJx'F=3x 
By  adding  x  to  both  members  to  complete  the  square,  we  have 

4F^-\.4jxrF+x=4x 

2F+Jx=zt:2jx 
Or,  2F=^x,  orSjx 

Restoring  the  value  of  P,  we  find 


1^  ROBINSON'S  SEQUEL. 


Whence,  Ay^ — 16jx=x,  or  9a;  (1) 

From  the  second  of  the  given  equations,  we  have 


>/8(y-V^)-4=(y-V^)+l 
Squaring,         Q(y-.Jx)—4=(y—Jxy+2(y—Jx)+l 

Whence,         (v^J^) '— 6(y— V^)  =  —5 
And  y—Jx—3=±2  (2) 

Taking  the  plus  sign        y — 6=  Jx  (3) 

Taking  the  minus  sign    y — 1=  Jx  (4) 

Substituting  the  values  of  Jx  and  x  taken  from  (3)  in  (1), 
we  have 

4y2_-16y+80=y2_i0y+25;  or,   V— 90y+225 

Whence,  y=^l2±l,EI^^  or  y^^^l^J^ 

^/3  5 

Taking  the  values  of  the  same  from  (4),  and  substituting,  as 
before,  we  have 

4y2— •16y+16=y2__2y+l,  and  9y^—lQy-\-9 

Whence,        .  y=3,  or  :?,  and    y=l±>/^ 
3  6 

Substituting  the  values  of  y  in  (3)  and  (4),  we  have  the  val- 
ues of  X. 


(13.)     Given 
and 


^x^—Uy—U     x^     ^fA 

5y+ ^ y-^^ 


to   find    the 
»•  values  of  x 

and  y. 

Multiply  the  first  equation  by  3,  transpose,  &c.,  and  we  have 


x^  ,2x Ix^.x^ y 

lSyY~^3y~4~^ 


V^!^1^14=(^-^_15y-14)-94 


Put      P=Jx^ — 16y— 14  ;  then  we  shall  have 
Whence,  P=  ^^g.  db  H  =  ^^ ,  or  — 9^ 


'»4ji. 


ALGEBRA.  107 

That  is,  a;2—15y— 14=100,  or  V//. 

Or,         x^  =  15y+U4;  and  x^  =  \5y+^^U^.  (1) 

The  second  equation  may  be  written  thus, 


■Mn)-^i.-*i 


8y  '  \  3   '  2/  >*  3y 

Uniting  the  fractions,  and 


x^,/4x+3y\_^  I4x+3y 
Sy"*  \      6      /       ^l     12y 
Dividing  every  term  by  2y,  and  we  have 

x^        /4x-\-3y\  _x  /^rg-f3yy 
T62^~^\    12y    /     2y\    12y    / 

For  the  sake  of  perspicuity,  put  P=  i  — X-Jl  j  ,  then 

16y2      2y    ^ 

By  evolution,  -^— P=0 

4y 

Whence,  ^L=P^=i^±?^ 

16y2  12y 

Clearing  of  fractions,         Zx^  =  \Qxy-\-\'2.y^ 

Whence,  Qar^— 48a;y=36y2 

Add   64y2   to  both  members,  to  complete  the  squares,  then 

9a;2__48a;y+64y2  =  100^2 

By  evolution,  3a; — 8y=  ±10y 

Whence,  a:=6y,  and  a;= — fy  (2) 

Substituting  the  first  of  these  values  of  x  in  equation  (1),  we 

have 

36y2_i5y=ii4 

By  adding  ||  to  both  members,  (Art.  99,  Algebra,)  we  shall 
have 

36y2_-16y+?.|  =  114+f|  =  Hi^ 
By  evolution,  Qy —  f  =  =h  V 

Whence,  y=2,  or  — 1|. 


108  ROBINSON'S   SEQUEL. 

These  values  put  in  the  first  of  equations  (2),  give 

a:=12,   or   — y. 
By  taking  the  second  set  of  equations  in  (1)  and  (2),  we  shall 
find  other  values  of  x  and  y. 


(14.)     Given    j  ^^y — \z=^x^y — \y^      /  to  find  the  values 

and     (a;2_3=a:2y 2  (x^_y\>^    )  oi  x  and  y. 

Ans.  ar=l.    y=4. 
Put    a;^=P,  and  y^=  Q,  and  we  have 

P3_3     ^PQ(^p_^Q)  (2) 

Now  put  P=tj  Q,  and  equation  ( 1 )  becomes 
(471^+1)  §«—16w(2'  =  16. 
Conceive  w  to  be  a  known  quantity,  then  the  last  equation  is 
quadratic,  and  a  solution  gives 

^3_4(2/i^+2^+l)_ 4^ ^__ 4 

But  from   (2),    ^^= - = - 

Put  the  two  values  of  Q^  equal,  and  put  n^ — n-\-l=E,     (3) 
Then    1_  =t-^  .     Whence,  2w  =  ^^ZI? .  (4) 

But  from  (3)  resolved  as  a  quadratic, 

2w=l=fc7(4i?— 3)  (6) 

From  (4)  and  (5),  2E  ±2i2^(4J!?— 3)=6i2— 3 

Or,  ±:2EJ  (4E—3)  =  4JS— 3 

Put  J(4E—3)=S.    . 

Then  '  S^:^2BS  =  0.     Or,         S(Szt2E)  =  0. 

This  last  equation  may  be  verified  by  taking  either  factor  equal 
to  zero  ;  and  as  the  first  factor  only  gives  a  rational  quantity,  we 
take  that  which  gives  i2=f. 

By  retracing,  we  easily  find  x  and  y. 


ALGEBRA.  109 

We  now  add  a  few  unwrought  examples  for  the  benefit  of  those 
who  may  wish  to  test  their  own  unaided  powers  in  these  difficult 
operations. 

None  of  these  that  follow  are  as  severe  as  many  of  the  prece- 
ding. 


(15.)     Given     (. /J-^~"^-f  ./-^^— =2  )   to  find  the  values 

and     (  a:2_i8=;c(4y— 9)  )   o^  ^  and  y. 

Ans.  x=6,  or  3. 
y==3,  or  f . 

(16.)     Given        (,+,)_^(,rz,T) ^- 

and     (  (a;2-f-3/)2_|_(.^_y)  =  2x(x^-^y)-{-50e 
to  find  the  values  of  x  and  y. 

Atis.  x=5,  or  — ^/. 
y=3,  or  ~|^. 

(17.)     Given    ^^+^^^^^24  1         _  ^^         . 

^      '  !    a; ^y     x-\-y     6  I  ^^  ^^"-  ^"®  values 

^        4ar2  \ 

and        V^— y+^=9^(^_^-)  of  ar  and  y. 

^W5.  ic=3,  or  Y  J  or  t\»  or  jf . 
y=2,  or — ^P;  or  I,  or — '^4^. 


(18.)     Given    W6V^+6Vy+W^=9-Wy  I    ^  fi„d  the 
and     (  a; — y=l2  J 

values  of  x  and  y. 

^Tis.  a;=16,  or  sjyuLP. 


y=4,  or  ^eV-'*, 


(19.)     Given    (x+JSy^—n+^x=7+22j-y^]    to  find  the 

,    i      73^71:^X7-^+2^  y  values   of 

and    j^  V  "^^    ^+^""^1::^  j   X  and  y. 


^?w.  ir=4.     y=2. 


110  ROBINSON'S  SEQUEL. 

(20.)     Given     j  a;^— 2/^=3  ) 

and     (  {x^+y*y+x'y''{x''—^'')^+x^-^^=32Q  \ 
to  find  the  values  of  x  and  y. 

Ans.  a;=s=  ±2,  or  ±J(—1). 
y=r  ±1,  or  ±2V(— 1). 

(21.)     Given    f  ^+J^+J_Jx-^^^Q9_  ^  ,^  g^^  the  val^ 
and    I      2 r~2_-.l^  [  ^^s  of  iT  and  y. 

^«*.  ar=9,  or  J^f^,  or  ^-f »,  or  16. 

y=4,  or  —  V>  or  —  V»  or  i. 


SECTION   IV. 

PROBLEMS  PRODUCING  QUADRATIC  EQUATIONS  CONTAINING 
MORE  THAN  ONE  UNKNOWN  QUANTITY. 

The  following  outlines  of  operations,  refer  to  problems  in  Rob- 
inson's Algebra,  Chapter  III,  page  183.  We  pass  on  to  the  sixth 
problem,  page  186,  and  only  include  those  which  serve  to  illus- 
trate brevity  and  elegance  in  operation. 

The  figures  in  parenthesis  refer  to  the  number  of  the  problem 
in  the  book. 

(6.)  Let  t  =s  the  time  (hours)  he  traveled,  and  r=  his  rate 
per  hour ;  then        r^=36  (1) 

But  if  r  becomes  (r+l),  t  must  become  (t — 3),  and  then 

{r+l){t-S)^36  (2) 

Or,  r/_|_jf_3r— 3=36 

ri  ac36 

^3(r+l) 
Hence,  ?-^+rs=12,  and    r=3. 

(7.)     Let  x=  the  number  of  children, 

and  y=  the  original  share  of  each. 
Then        a:y=4680O  (1) 


ALGEBRA.  Ill 

(x—2)  (y+1950)=46800  (2) 

ajy+1950a;—2y— 2- 1950=46800 
1950(a;— 2)  =  2y 

Or,  975(x — 2)z=xi/=i46809 

By  division,     x^ — 2a;=48 x=B» 

(8.)     Let  x=  the  number  of  pieces. 
Then  =  the  cost  of  each  piece. 

X 

48a:— 51^=675 

X 

4Qx^—675x=675 
16x^—225x=225. 


(9.)     Let  xz=  the  purchase  money. 

Then    12^=  the  cost,  and    390—^^^'^=  his  whole  gain. 
100  100  ^ 

Then    12^   :  390-12^  :   :  100  :  ^ 
100  100  12 

Product  of  extremes  and  means, 

^^^=39000— 104a? 
300 

—  =3000— 8a: 
300 


Put  a  =  300  and  divide  by  2  ;  then 


— =i5a — 4a: 
a 

x^-\-4ax=5a^ 

a:2_|_4aa:+4a2=9a2 

a:-|-2a=3a a;=a=»300. 

(10.)     Put  a;+y=  the  greater  part, 

and  X — y=  the  less  part. 
Then  2a:=60,  a:=30,  and  ar^—y 2  =  704. 

(11.)    Let  a;=  the  cost ;  then  89 — a:=  the  whole  gain. 

X  :  39— a;  :  :  100  :  x.  Ans.  ar=10. 


112  ROBINSON'S  SEQUEL. 

(12.)     Let  (x — 20)  =  the  number  of  persons  relieved  by  A. 

Then  x-\-20  =  the  number  of  persons  relieved  by  B. 

1200   ,  c       1200 
+5=. 


a;-}-20  X — 20 

Divide  by   5,  and  put  a=240  ;  then 

'  «         fl:  « 


a:+20  a;— 20 

aa;— 20a+a;2— 400=aa?4-20a 

a:2=40a-i-400=40(a+10)=40-260 

Or,  a;2=400-25 a?=20-5=100. 

Hence  80  is  ^'s  number,  and  120  A's. 

(13.)     Let  x=  the  price  of  a  dozen  sherry 
and  y=  the  price  of    a  dozen  claret. 

7a:+12y=50  (1) 

— =  the  number  of  dozen  of  sherry  for  10£. 

X 
n 

-=  the  number  of  dozen  of  claret  for  6j£. 


(2) 


By  substitution,  _Z2^4-12y=50 

70y-|-36y2  -f72y  =  1 50y+60  •  6 

36y2_8y=300 

92/2— 2y=75.     Hence, y=3. 

(14.)     Let  19a:=  the  whole  journey. 
Then  x=  £'s  days,  also  his  rate  per  day. 
Or    x^  =  £'s  distance. 
Also,    7a;-j|-32=  A's  distance. 

a:2_|_7a:_j.32=19a; 

x^—12x=—S2. 

Hence, a?=8  or  4. 

And 19ir=162  or  76. 


y 

Then 

12=3+? 
X            y 

Or, 

^       10         \Qy 
36     3y+6 

y 

ALGEBRA. 


113 


If  we  put  X  for  the  whole  journey,  we  shall  obtain  the  13th 
equation,  (Art.  104.) 

(16.)     Leta?=  the  bushels  of  wheat, 

and  ar-4-16=  the  bushels  of  barley. 
24__    24,  1 
~x      .r+16'^4 

24a;+16  •  24=24a:+^J±l?.^ 

a;2-[-l6a;=16- 96=16- 16-6 
Put  2o=16.     Then  2a-2a-6=24a2 

ar+a=  ±:ba .a;=4a=32. 


(16.)     A  put  in  4  horses,  and  B  put  in  x  horses. 

18 
Then    — =  the  rate  per  head. 

X 


Hence, 


4»18 

X 

4»20 
x+2' 
4-18     4-20 


|-18=  the  price  of  the  pasture. 
j-20=  the  price  of  the  pasture. 


X 

36^ 

X 


x-\-2  ' 

:J^+1. 

x+2^ 


x=6. 


;» 


(17.)     Let  4x=  the  price  per  yard, 

and  9x=  the  number  of  yards. 
36;r2^324 


x=3. 


(18.)     Let  10x-\-y=  the  number. 


Then 

And 

From  (1), 
From  (2), 

By  division, 
8 


xy 
10a:+y4-27=10y+a; 
10a:=(2a; — \)y 
x+3=y 

i^=2^-l 
a:+3 


(1) 
(2) 


/  ^ 


114       '  ROBINSON'S  SEQUEL. 

2a;3_-5a;=3 «=3- 

(19.)     Let  (a;— y),  x,  and  (a;+y— 6),  represent  the  numbers. 

Then  3a?— 6=33,  or  x=lS. 

(ic— y)2=a;3__2a?y+y2 

x^=x^ 
(x^y—6y==x^  +2xy+y^-'12x^l2y+S6 
3a;2_|^22/2— 12^— 122/+36=441 
By  subtracting  the  value  of  3a;2— 12a;+36,  we  have 

2y2 — 122/=64.     Hence, ir*^. 

(26.)  Let  a;+y=  the  greater,  and  x—y=  the  less. 

Then  (a;2— y2)(2a:2+22/2)=1248  (1) 

Or,  ,     a;4_3^4^624 

Also,  4xy=:20  (2) 

^  6        .     626 

Whence,  y=-.    y*=-i- 

a:*-^-=624 
a;* 
a;8— 624a;4  =626.     Put  2a=624. 
Then  x'^—2ax''+a^=a^+2a-\-l 

x'—a=  ±(a+l) 
Whence,  a;4=2a+l=625.     a;2=d=26. 

a;=6,  or  — 6. 

From  (2),  y=l- 

(27.)     Let  x=  A's  stock.    a=1000.    Then  a-^=  B's  stock. 

Observe  that  780=  the  whole  gain. 

Then  9a;+(6a— 6a;)=6a+3a;  :  9x  :  :  780  :  1140— a:. 

Or,  2a+x  :  Sx  :  :  780  :   1140— a:. 

This  proportion  will  produce  a  laborious  equation  to  work 
through.  Therefore  we  will  try  2x  to  represent  ^'s  stock  ;  then 
9-2a;=18ic.     (a— 2a;)6=6a— 12a?. 

18a;+(6a-12ar)=6a+6a;  :  18a?  :  :  780  :  1140— 2a?. 

Reducing,  gives  us 

a^x  :  3a?  :  :  390  :  670— a?. 
670a— aa?+670a?—a?2  =  l  170a:. 


ALGEBRA.  116 

Whence,  a?2+1600a^=  570000. 

a;24-1600:c+(800)2  =  1210000. 
a:4-800=1100. 

a;=300.     2^=600,  ^'s  stock. 
When  the  Algebra  was  first  published,  the  6  months  in  the 
problem  was  printed  8  months,  by  mistake.     How  could  we  dis- 
cover that  mistake  ? 

We  look  at  the  answer  and  see  that  the  numbers  600  and  400, 
make  the  stated  sum  1000  ;  therefore  we  will  assume  that  these 
three  numbers  are  correct.  We  will  now  take  m  to  represent  9, 
and  n  to  represent  ^'s  time.  Then  the  preceding  proportion 
becomes 

^mx—^nx-\-7ia  :  ^mx  :  :  780  :   1140—22:. 
Also,      ^mx — 9,nx-\-na  :  na—^nx  :   :  780  :  640-(-2a; — a. 
Now  give  to  X  its  value  300,  and  to  a  its  value  1000,  and  these 
proportions  will  give  m=9,  and  ?^=6. 

(28.)     Let  «2_.  half  the  number  in  the  first 

Then  SLx^is=^  the  number  in  the  first. 

And  4a;-|-4=  the  number  in  the  second/ 

3(2a;2-j-4a:-[-4)=  the  number  in  the  third, 
3(.'c2-[-2a!-f  2)-|-10=s  the  number  in  the  fourth. 

Sum,  ll(*'2-|-2a'+2)+10=!121,  the  given  sum. 
Whence,  a;  ^-j- 2^+2=  101. 

Or,  a-'2+2.r-f-l  =  100. 

By  evolution,  x-^\=  ±10,  or  .r=9,  for  the  minus  sign  will 
not  apply.     Then  22'2  =  162,  the  number  in  the  first, 

(31.)     Let  a:=  the  greater  of  the  two  numbers, 

and  y=  the  less. 
Then  per  conditions,     xy-s^^x"^ — y^  (1) 

And  a-'2-|-2/2_^3_^3  ^gj 

From  ( 1 ),  x^  — ^y=y^  • 

Conceive  y  a  known  quantity  and  complete  the  square  thus  ; 

4a;2_4y.a:_j-y2_5^2 

2.r— 2/=  ztj5'7/ 


116  ROBINSON'S  SEQUEL. 

Or,  22:=(lrhV5)y.     Let  (l±V5)=a. 

Then  x=^  (3) 

2  . 

Let  this  value  of  x  be  substituted  in  (2),  and  we  have 

4    ^^  8       ^ 

Dividing  by  y*  and  clearing  of  fractions,  and 
2a2+8=(a3— 8)y 

Whence,  y= ^— 

Buta2=6±275.     aS^^iedrS^S.     Therefore, 

^       8±8V5      2\1±V5/  ^ 

This  last  operation  may  not  be  obvious  to  some  ;  it  will  be  seen 
by  multiplying  (1±V^)'  ^7  V^'  ^^^*  ^^'  ^^^^  numerator  in  paren- 
thesis is  ^5  times  the  denominator. 

To  find  X  we  must  simply  multiply  y  by  |a,  see  (3)  ;  that  is, 
ar=Kl±V5)>/5=i(V5±5). 


The  following  are  not  in  Robinson''s  Algebra,  but  selected  from 
every  source, — mostly  from  Bland^s  Problems. 

(1.)     The  swn  of  two  nwmhers  is  2,  and  the  sum  of  their  fifth 
powers  is  32.      What  are  the  numbers.^ 
Let  x=  one  number,  and  y=  the  other. 
Then  x+z/=2  (1) 

And  a:5+y'=32  (2) 

As  the  5th  power  of  2.  is  32,  therefore 
(x^y)^=x^-\-y^ 
That  is,  x^-^-dx'y+lOx^^j^+lOx'^y^-^-Bxy^+y^^x^-^-y^ 
Or,  5x*y+10x^y''+10x''y^+5xy^=0  ' 

By  division,         x^ -\-2x^ y-\-2xy^ -]-y^  =0 
That  is,  x^-\-y^-{-2xy(x-\-y)=0 

Dividing  by  {x-\-y),  and  we  have 

g.2_^_^y2_^2xy=0 


ALGEBRA,  117 

Or,  x'+x7/+y^=0 

From  (1),  x^+^xi/+y^=4 

By  subtraction,  a;y  =4  (3) 

Multiplying  (1)  by  «,  gives    x'^-\-xy=^^ 

That  is,  x^—2x=z—4 

Whence,  x=l±^—3 


Then  y=lqr^— 3 

Here  we  have  obtained  two  expressions,  the  sum  of  whose  5th 
powers  is  32,  but  not  two  numbers. 

We  have  not  so  clear  an  idea  of  (1=1=^ — ^)>  ^^  ^^  ^^^^  ^^  ^ 
itself.  If  we  compare  equations  (1)  and  (3),  we  shall  perceive  an 
impossibility/;  for  two  numbers  whose  sum  is  only  2,  can  never 
make  a  product  of  4.  In  the  same  manner  when  a  sum  is  but 
2,  the  sum  of  the  5th  powers  of  any  two  of  its  parts,  can  never 
make  32. 

To  test  our  quantities,  we  will  verify  (2)  with  them.  To  save 
trouble,  we  will  put  a=jj — 3,  then  a^  =  — 3,  a'^=9. 

y5  =  (l— a)5  =  l— 5a4-10a^— 10a^+5a^— gs 
aj5-|-y5  =2-f-20a2_j_l0a*  =2— 60+90=32. 

(2.)  The  fore  wheels  of  a  carriage  make  6  revolziiions  mx>re  than 
the  hind  wheels  in  going  120  yards  ;  but  if  the  periphery  of  each 
wheel  be  increased  by  one  yard,  then  the  fore-wheels  will  make  only  4 
revolutions  more  than  the  hind  wheels,  in  running  over  the  same  dis- 
tance.    Required  the  circumference  of  each  wheel  ? 

Ans.  Fore  wheels,  4,  hind  wheels  5  yards. 
Let  ir=  the  yards  in  the  circumference  of  the  larger  wheels, 
and  y=  the  jrards  in  the  circumference  of  the  smaller.  ' 

Put  a=120. 

Then  per  question,     -=- — 6.  '  (1) 

X    y 

And  _JL.=_^_4  (2) 

^+1     y-f  1  ^   ^ 

Clearing  of  fractions,     ay=iax — ^xy.  (3) 

ay-\-a-=ax-\-a — ^xy — ^x — Ay — 4.  (4) 


ria  ROBINSON'S  SEQUEL. 

Suppressing  a  in  both  members  of  (4),  and  then  subtracting  it 
from  (3),  we  have 

0=— 2xy+4;r+4y+4.  (6) 

From  (3),         :r=  -^=  J^^.  =  ^^ 
a—Qy     12U— 6y     20— y 

From  (6),         0:=?^:? 

Therefore,  Hd=J^ 

y— 2     20— y 
20y— y  2  _j_2o_y  =  1  Oj/2  _20y 

Whence,  ll2/2_39y=20. 

If  we  work  out  this  quadratic,  we  shall  find  y  =  4 ;  but  the 
operation  would  be  a  little  troublesome,  because  the  numbers  are 
prime  to  each  other. 

In  cases  like  these,  when  a  practical  operator  is  only  in  pursuit 
of  results,  he  looks  at  the  absolute  term,  (in  this  example,  20), 
and  observes  its  factors,  2,  10,  4,  5,  and  conceives  y  to  represent 
one  of  them  ;  and  if  it  verifies  the  equation,  then  y  is  really  that 
factor. 

I  will  now  conceive  y  to  be  4,  and  divide  the  first  member  by 
y,  the  second  by  4  ;  then 

'  lly— 39=5 

Or,  lly=44,  or  2/=.4. 

Therefore,  as  this  supposition  verifies  the  equation,  the  suppo-' 
sition  itself  is  truth. 

Now  let. us  suppose  y  to  be  6  ;  then  operate  as  before,  and 
lly— 39=4 
ny=43 

Now  as  y  does  not  come  out  equal  to  5,  the  supposition  was  not 

true. 

■c  1,  2O2/       20-4     - 

l*or  X,  we  have   x=. ^-= =5. 

20— y       16 

(3.)  A  and  B  engaged  to  reap  afield  for  ^24  ;  and  as  A  could 
reap  it  alone  in  9  days,  they  promised  to  complete  it  in  5  days. 
Finding,  however,  that  they  were  unable  to  finish  it,  they  called  in  0 
to  assist  them  the  last  two  days,  in  consequence  of  which,  B  received 


^ 


ALGEBRA.  119 

$1  less  than  he  otherwise  would  have  done.     In  what  time  could  B  (yr 

C  alone  have  reaped  the  field? 
Let  x=  the  number  of  days  in  which  B  ffould  reap  the  field, 
and  y=  the  number  of  days  in  which  C  could  reap  it. 
As  A  could  do  it  in  9  days,  for  one  day's  work  he  should  have 

I  of  the  money  ;  and  as  B  could  do  it  in  x  days,  for  one  day's 

work  he  should  have  -  of  the  money.      A  and  B  then  working 

X 

together  one  day  would  do  --[--  o^  the  work,  and  in  5  days  they 

y     X 

would  do  s/'l+i^  :  -  :  :  24  :  ?1?1=  the  number  of  dollars  B 
\9     x/     X  rc+9 

would  have  received  had  (7  not  been  called  in. 

But  as  B  can  reap  the  field  in  x  days,  for  one  day's  work  he 

24                                                         6*24 
should  have  —  dollars,  and  for  five  day's  work, dollars,  the 

X  X 

sum  he  did  receive. 

Therefore,       — =:1 

2^+9        X 

Whence,         x''—'^lx=:^\m<^. 

Here,  as  we  are  only  in  pursuit  of  results,  we  try  Inspection. 
We  perceive  that  10  for  the  value  of  x  would  not  be  large  enough, 
and  20,  too  large  ;  and  as  1080  terminates  in  a  cipher,  we  will 
try  dividing  by  1 5  ;  then 

«— 87=— 72.     Whence,  x=\b.  Am, 

Also,  a;=72  ;  but  this  will  not  apply  to  the  problem. 

Again,  as  A  could  do  the  work  in  9  days,  for  one  day's  work 

he  should  have  —  dollars,  and  for  5  day's  work,  — dollars. 

9  "^9 

5*24  2*24 

B  should  have dollars,  and  C,  dollars,  and  the  sum 

15  '  '     ^ 

to  the  three  is  24  ;  therefore, 

5-24  ,  5-24  ,  2-24 


=24 


9      •     15     '      ^ 
Or,  |-j_^_|_?=i.     Whence,  y=18,  Ans, 


^ 


120  ROBINSON'S  SEQUEL. 

(4.)  Bacchus  caught  Silenus  asleep  hy  the  side  of  a  full  cash, 
and  seized  the  opportunity  of  drinking,  which  he  continued,  for  two-, 
thirds  of  the  time  Silenus  would  have  taken  to  empty  the  whole  cask. 
After  that,  Silenus  awoke  and  drank  what  Bacchus  left.  Had  they 
both  drank  together,  it  would  have  been  emptied  two  hours  sooner,  and 
Bacchus  would  have  drank  only  half  what  he  left  Silenus.  Required 
the  time  in  which  each  would  have  emptied  the  cask  sepa/rately. 

Ans.  Bacchus  in  6  hours,  and  Silenus  in  3  hours. 

Let  a=  the  volume  of  the  cask. 

a;=  the  time  Bacchus  would  require  to  drink  it  alone. 
y=  the  time  Silenus  would  require  to  drink  it  alone. 

Then  -=  the  volume  Bacchus  drank  per  hour. 
And    _=  the  volume  Silenus  drank  per  hour. 

y 

- .  -^=  the  volume  Bacchus  drank ;  then 
X    3 

(a — -^jz=  the  quantity  left  to  Silenus  ;  and  this  quantity  divid- 
ed by  the  volume  Silenus  drank  per  hour,  will  give  the  hours  he 
employed  in  drinking. 

That  is     (a—?^\l ,  or  (y^?^\  =  the  time  Silenus  drank. 

Had  they  both  drank  together,     (J^)   would  express  the 

\x-\-y/ 


I 


time. 

Now  by  the  given  conditions. 

3^*       3x     »+y 

(1) 

And                 ("    ''VY-_''y 
\2     Sx/a    «+y 

(2) 

Reducing  (2).  and     l    ^-/^ 

Or,                            3x'—2y'=5xy 

9a»— l&ry=6y»          . 

2 

Whence,         x=2y. 
This  vahie  put  in  (1),  gives 


ALGEBRA.  Ill 


3~^     3      3~ 


(6.)  A  Banker  has  two  hinds  of  money  ;  it  takes  a  pieces  of  the 
Jirst  to  make  a  crown,  and  b  pieces  of  the  second  to  make  the  same 
sum.  Some  one  offers  him  a  crown  for  c  pieces  :  how  many  of  each 
kind  shall  he  take  ? 

Ans.  Of  the  first  kind  l       ^J^   of  the  second,    i 4-- 

(6— a)  (a— 6) 

This  problem  is  more  of  a  puzzle  than  most  others,  yet  it  is  a 
fair  scientific  question. 

Let  a;=  the  number  of  pieces  of  the  kind  a, 
and  y=  the  number  of  pieces  of  the  kind  b. 
Then  x-\-y=c.  (1) 

As  a  pieces  are  worth  1  crowiji,  one  piece   is  worth  -,  and  x 

a 

pieces  are  worth  — 
a 

By  a  parity  of  reasoning,  y  pieces  of  the  second  are  worth  - 
and  the  worth  of  both  together  is  just  1  crown  ;  therefore, 

?+|=l  (2) 

a     0 

Whence,  hx-\-ay=^ah 

From  (1),  hx-\-by=bc 

By  subtraction,     (a — b)y=(a — c)b.     y=S^^ZZL. 

a — 5 


In 'like  manner  we  find  x=A ^'. 


Itt  ROBINSON'S  SEQUEL. 

(6.)  A  and  B  traveled  on  the  same  road,  arid  at  the  same  time, 
from  Huntington  to  London.  At  the  bOth  mile  stone  from  London, 
A  overtook  a  drove  of  geese  which  were  proceeding  at  the  rate  of  3 
miles  in  2  hours;  and  iwo  hours  afterwards,  met  a  stage  wagon, 
which  was  moving  at  the  rate  of  9  miles  in  4  hours.  B  overtook  the 
same  drove  of  geese  at  the  Abth  mile  stone,  and  met  the  same  stage 
wagon  exactly  forty  minutes  before  he  came  to  the  ^\st  mile  stone. 
Where  was  B  when  A  reached  London  ? 

Ans.  25  miles  from  London. 

Let  x=.  the  rate  which  A  and  B  traveled  per  hour. 
Then  50 — 9,x=  the  distance  from  London  where  A  met  the  stage. 

m     h         m 

3  :  2  :  :  5  :  y  =  ^li®  hours  required  for  the  geese  to  travel 
6  miles. 

Then  when  B  was  45  miles  from  London,  A  must  have  been 

50 — )  miles  from  the  same  place,  and  the  distance  between 


3  / 


( 

the  two  travelers  must  have  been  i — 5  \  miles,  and  the  val- 
ue of  this  expressson  is  the  answer  demanded. 

Now  let  t  be  the  hours  elapsed  between  the  times  that  A  and  B 
met  the  stage. 

The  motion  per  hour  for  the  stage  was  f  miles. 

B  met  the  stage  /  314-— )  miles  from  London  ;  but  A  met  it 
before,  nearer  to  London  by  —  miles.     That  is,  A  met  the  stage 

(31 -4-— — .  J  miles  from  London.     We  have  before  determined 
^3       4/ 

that  A  met  the  stage  (50 — 2x)  miles  from  London;  therefore, 

31+?^— -5^=50— 2a:  (1) 

^3       4  ^   ' 

Now  after  A  met  the  stage  he  traveled  in  one  direction,  and  the 

stage  in  another  for  t  hours,  before  the  stage  met  B,     Then  their 

distance  asunder  must  have  been  l-^-\-tx\.     But  the  distance 


ALGEBRA.  123 

the  two  travelers  are  asunder,  has  been  expressed  by  ( — 5  ) 

Therefore,  ^+to=i— — 5  (2) 


From  (1),  /= 

.32.-228      j.^^^    ^^    ^_40.-60 
27                       '    '         27+ 12a; 

Therefore, 

3ar— 228_40a;— 60 
27           27+12a; 

Or, 

8a;— 57__10a;— 15 

•  9  9+4aj 

Clearing  of  fractions, 

32a;2_228if+72a;— 513=90a;— 135 
Or,  16a;2— 1232^=189  V 

Here  the  obvious  whole  number  factors  of  189  are  3,  9,  and 
21  ;  and  as  we  are  only  in  pursuit  of  results,  we  will  try  one 
or  two  of  them.  21  we  perceive  at  once  is  too  large,  therefore, 
try  9  ;  then 

16a;— 123=21 

16a;=144,  or  a;=9,  a  true  result. 

Now  because  a;=9,     — f — 6=25,  the  answer  to  the  question. 


SECTION  v. 

PROBLEMS   IN"  PROPORTION",  AND  IN"  ARITHMETICAL, 
GEOMETRICAL  AND  HARMONICAL  PROGRESSION". 

The  problems  contained  in  Robinson's  Algebra  are  not  written 
out ;  they  are  only  referred  to  by  article,  and  number  of  the  prob- 
lem, and  a  mere  outline  of  the  solution  indicated. 

We  commence  with  Art.  117,  example  3. 

(3.)  Let  X — 3y,  x — yy  a;+y,  and  x-\-%y  represent  the  numbers ; 
then  2y=4. 


124  ROBINSON'S  SEQUEL. 

The  product  of  the  1st  and  4th,  is 

a;2_9y2  .  of  the  2d  and  3d,  is  (x^—y*). 


.a;4_9^22^2 

jc4_ioa:2y2^9y*  =  176985 

9y*=       144 

a;4_40a;3  =176841 

(4.)     The  same  notation  a^  in  the  last  example. 
2;r=8.     x=4.     x^—y^z=15. 

(6.)     Let  n=  the  number  of  days. 
Then  L=\+{n—\)\=n. 

S=^{\-\-n) ^n=i  the  whole  distance. 
Also,  (n — 6)15        =  the  whole  distance. 

«2__29^^_I30 w=9or20. 

9—6=3.     20—6=14. 

(6.)     The  first  day  he  must  pay  l+^;  i  representing  the  in- 
terest of  one  dollar  for  one  day. 

First  day,  \-\-    i. 

2d  day,  1+  2e. 

3d  day,  l^-  3i. 

Last  day,  1+60J. 

(2-|-6U)30=  the  whole  sum  to  be  paid  ;  but  as  this  sum  is  to 
be  paid  in  60  equal  payments,  each  payment  must  be 

^^  1^ ^-=  Ans.  81  and  |  of  a  cent,  nearly. 

(7.)  Let  X — 3y,  x—y,  x-\-y,  and  x-\-^y  represent  the  numbers ; 
then  2;r2+18y2=50 

2a;2-|-  2y^=34 

16y2  =  i6 y=l. 


ALGEBRA.  t26 

GEOMETRICAL    PROGRESSION-   AIND   HARMONICAL 
PROPORTION. 

(Art.  124.) 

(1.)     Let  X  represent  the  mean  sought.  / 

18 

(2.)     Let  x=  the  number  sought.     Then,  by  harmonical  pro- 
portion                    234  :  X  :   :.  90  :   144— x 
90a;=234-144— 234x 
324a:=234- 144.     Hence, ar=104. 

(3.)     Let  x=  the  number  sought. 

Then  24  :  a?  :  :  8  :  4—x 

Or,  •    ,  3  :  a;  :  :  1   :  4—x x=3. 

(4.)     Let  x=  the  second. 

Then  16  :  2  :  :   16— a:  :  1 a;=8, 

(6.)     Let  x=  the  first  number,  and  y=  the  ratio. 
Then  x+xy-\-xy^  =nO  (1) 

xy^—xz=i90  (2) 

By  subtraction,,  2a;4-^y=120,  or  .t= 

90 
From  (2),  we  hare  x= 

-— =— L  ,   or   4y2— 3y=10 y=9L 

(5.)     Let  X,  xy,  xy^ ,  and  xy^  represent  the  numbers. 
Then  ^y3     __  y2   _4 

xy-^-xy""      1+y     3 
From  this  equation  we  perceive  at  once  that  y=2 ;  then 

a:+2a;+4a:-f  8ar  =  1 5ar=30 ar!=  2. 


t28  KOBINSON^S  SEQUEL. 

(6.)     Let  X,  xy,  xy^,  and  xy^  represent  the  numbers^ 

a?+a-2/2  =  i48  (1) 

a'y+3ry3=888  (2) 

Or,  <l+y2)  =  4-37  (3) 

^y(l+y')  =  4-222  (4) 

Divide  (4)  by  (3),  and, y=6. 

(7.)     Let  x,  J(xy),  and  y  represent  the  numbers  ;  then 
«^+>/{^)+y=-14  (1) 

^And  a;2-}-a;2/+y2      ^34  (2)      ' 

Put  a;-|-3/=5,  and  J{xy)=^p; 

Then  a;2-|^5^-|-y^="^'^' — i^^»    and  equations  (1)  and  (2) 

become  s-|-jys=i4  (3) 

s2_jo2^84  (4) 

Divide  (4)  by  (3),  and  we  have         s^—psnQ  (6) 

Add  (3)  to  (5),  and  divide  by  2,  and   5fc=10. 

Hence,  . . .  ^ » » v . , . .  j9s=4. 

(8.)     Let  X,  xy,  xy^ ,  and  a-y^  represent  the  numbers  ;  then 
xy^-^xy=^9.A 
xy^+x  :  xy^-{-xy  :  :  7  :  3 
Or,  y^-\-l     :       e/^+y  :   :  7  :  3 

Divide  the  first  couplet  by  (y-^l),  and  we  have 
y'—y+l   :         y         :  :  7  :  3 
3y2_32^_|_3_.7y^  ^31-  32/2— .10y=t— 3. 

From  this  equation  we  have  y=3,  the  ratio. 

(9.)     Let  X,  xy,  xy^ ,  and  xy^  represent  the  numbers  \ 
Then  ar(l+y+y2+y3)=:y-|-l 

And  «:=rV-  Put  (y+l)=^. 

Then  j\{A+Ay^)=A 

A-^-Ay^^lOA.  Ay^==9A,  or » . . .  .y=3. 

Hence,  xV>  tV>  <^<^-  ^^^  ^^^^^  numbers. 


ALGEBRA.  tf7 

(10.)     Let  X,  — —y  and  y  represent  the  numbers  ;  then 

^+^+y=26  (1) 

And  a?y=72 

Put  a:-j-y=* ;  then  equation  ( 1 )  becomes 

144.  ^ 

s+ilZ=26,  or  s2_26s=— 144 s=.tl8. 

s 

(11.)     Let  a?,  a;y,  and  xy^  represent  the  numbers  ; 

Then  x^y^^2\Q  (0    '^k 

a;2_f-a;2y4^328  (2)      ^ 

From  (1)  <Py=6,    or  x^= — 

From  (2) 


1+y^ 


36_  328      ^^9^    82         , 

9y  4___82y2  =  —9.     Hence, ys=3i 

(12.)     Let  Xt  Jxy,  and  y  represent  the  numbers  :  then 

^+>/^+y=i3^  (1) 

{x+y)Jxy=^Q_  (2) 

a;+y=13— 7a;y  •  (3) 

_        30  _ 

13 — Jxy=—=r-     Hence  J xy =3. 
Jxy 

(13.)     Let  X,  — ^,  and  y  represent  the  numbers  ;  then 

^+y=i8  (1) 

?^!=.676  (2) 

18  ^   ' 

^1=M.     xy^l^  (3) 

o 

From  (1)  and  (3),  we  find  x  and  y. 


128  ROBINSON'S  SEQUEL. 

(14.)     Let  X,  xy,  and  xy^  represent  the  numbers  ;  then 
(^xy^ — xy)  (xy — x)  are  the  first  diflferences,  and 

xy^ — 2xy-\-x=  6 

xy'-\-  xy-\-x=42 
Difference,  Sxy      =36 xy=l2 

(15.)     Let  a;,       ^  ,  and  y  represent  the  numbers.     If  y  is 
x+y 

supposed  greater  than  x,  then  (  y —   ^^  j  i    ^^ — x  j    are   the 

1st  differences,  and  y — . — ^-Ua;=2,  the  2d  differences.- 
x+y 

ajy=72.  Put  {x+y)=s; 

Then       .  s— l_if=2 

s 

s2_25+l=289 
«— 1=17.         s=X'\-y=lS. 

(17.)     Let  a;^,  xy,  and  y^  represent  the  numbers  ;  then 
a;a_|_a;y+y2=:3i,  and  ic 2+2/2  =26. 

(18.)    Let  X,  xy,  xy^ ,  xy^,  xy^^  and  xy^^  represent  the  num- 
bers.    Then,  by  the  conditions,  we  have 

x-\-xy-\-xy^-\-xy^-\-xy'^-\-xy^  =  lB9=a  (1) 

And  xy-\-xy'^=54=:b  (2) 

But  equation  ( 1 )  may  be  put  into  this  form 

( 1+y+y  ^  )^+(  1 +y+y' >y^  =« 

Or,  x+xy^=    -^— 

Multiply  this  last  equation  by  y,  and  its  first  member  will  be 
the   same  as   the  first   member  of  equation   (2).      Therefore, 
^^ =  6  :  a  quadratic  from  which  we  obtain  y,  the  ratio. 

(19.)     Take  the  same  notation  as  for  (18)  ;  then  we  have 
(a:-fary)+(ir+a:2/)y^  =  189— 36=:153=a.  (1) 


ALGEBRA. 
And  (x+xy)y''=36=b. 

Divide  (1)  by  (2),  and  we  have 

1+2^^^153^51^     Hence. 


y' 


36      12 


(2) 


129 


r=2. 


CHAPTER  III.— PROPORTION. 


(5.)     Let  X  and  y  represent  the  numbers  ;  then 


x—y  :  x-{-y 

x-\-y  :  xy 

From  the  first,  2a;  :  2y 

ISy  .  Il2^ 

7  *  7 

y  :  lly^ 


2 
18 
11 

18 

1 


9 

77 
7,    or  x=yy, 

77 

77.    y=7. 


(6.)     Let  a:  and  y  represent  the  numbers. 

a;+4     :     y+4     :    :     3     :     4  (1) 

ar_4     :     y— 4     :    :     1     :     4  (2) 

From  (2)  we  have  4x — 16=y — 4,  or  y=4x — 12.     This  value 
of  y  put  in  (1),  gives 

a;+4     :     4x—S     :    :     3     :     4 
ar-|-4     :       x—2     :    :     3     :     1 

a;_^4=3ar— 6 a;=6. 

(7.)     Let  X  and  y  represent  the  numbers  ; 

Then  x-\-y=l6 

And  xy      :     ar^+y^     :    :     15     :     34 

Double  the  first  and  third  terms,  then  add  and  subtract,  (  The- 


orem  4),  and     2a?y 

x^+y' 

\    :     30 

:     34 

x'+^y+y'    : 

x^—^y+y^ 

:     64 

4 

^+y 

x—y 

:       8 

:       2 

16 

x—y 

:       4 

1 

Or. 


«— y=4. 


180  ROBINSON'S  SEQUEL. 

(8.)     Let  x=  the  gallons  of  rum. 
And  j/=  the  gallons  of  brandy, 

x—1/     :    y     :    :     100     :     X 
B       X — y     :    X     :    :         4     :    y 

Product,  (x—yY    \    xy    \    \     400     :  xy 

Dividing  the  second  and  fourth  by  xy,  and 

(x—yY     :     1     :    :     400     :  1 
x—y        :     1     :    :       20     :     1,    or    x—y^10. 


(9.)     Let  x-\-y=i  the  greaternumber, 


And 

X — y=.  the  less. 

Then 

a;2_y2=:=320. 

(ar+y  )  3  =;^3_|.3^2y^3^y2  _j.y  3 

(ar_y)3  _^3_3^2y_j_3^2/2--3/^ 

(1) 


6a;2y-|-2y3==  diflf.  of  the  cubes. 
2y=  difference.         Cube  of  (2y)=8y3 
6a;2y-|-2y3     :     Sy^     :    :     61     :     1 

Sar'-f-y^— 244y2.         3^.2^2432^2^ 

This  value  of  x'^  put  in  equation  (1),  gives 

80y2=320,  or y=2. 


We  now  give  additional  problems. 

(1.)    The  sum  of  four  whole  numbers  in  arithmetical  progression 
is  20,  and  the  sum  of  their  reciprocals  is  |f-.     What  are  the  numbers? 

Let  X — 3y,  x — y,  ar-f-y,  and  x-\-3y  be  the  numbers  ; 
Then  4a;=20,  and  x=5. 

Affam  + -4- + = — 

^  x—Sy    x—y  '  x-\-y    x-\-3y     24 

Uniting  the  1st  and  4th,  and  the  2d  and  3d,  we  have 


ALGEBRA.  131 

2x        ,       2a:         25 


Dividing  by  x=5,  and  then  clearing  of  fractions,  reduces  the 
equation  to     2x''—'2y''+2x^—lQi/^z=^^(x^—9y'')  {x^'—y''). 

96^2_43o^2^5^4_50a;2y2_j_45y4 

Putting  the  value  of  a;^  in  the  2d  member,  we  have 

96^-2_480^- ==  1 25;i;2  —  1 230y2  ^46y4 
Whence,  0=292;-— 770y--|-45y^ 

Dividing  by  5,       0=29a?— 154y2_j_9y4 
Or,  0=145— 154y2_j_9y4 

Here  the  sum  of  the  coefficients  is  the  same  in  both  members  ; 
therefore  one  of  the  values  of  the  unknown  quantity  is  1 ;  and  as 
y=l  answers  the  conditions  of  the  problem,  we  are  not  required 
to  find  the  other  roots. 

Hence  the  numbers  are  2,  4,  6,  and  8. 

(2.)  Tke  sum  of  six  numbers  in  arUhm^tical  2>rog7'ession  is  33, 
und  the  sum  of  their  squares  is  199.      What  are  tke  numbers  ? 

Ans.  3,  4,   6,  6,   7,  and  8. 

Let  [x — y)  represent  the  third  term,  and  {x-\-y)  the  fourth 
term ;  then  2y  will  be  the  common  difference,  and 

{x—Sy),  (^•— 3y),  {x—y),  {x-\-y),  {x-\-2>y),  and  {x+oy) 
will  represent  the  numbers. 

Then  6.r=33,  or  2.i  =  ll.  (1) 

And  6a:2-|-7Q?/2zz=199.  (2) 

From  (1),         4a.'2  =  121,  or  12^-2c=363, 
Double  (2),  and  write  363  for  \9^x^ ,  then  we  have 
363+140^2=398 
140y2=:35;        y=J-. 

(3.)  Find  four  numbers  in  proportion  such  tk^U  their  sum  shall 
ie  20,  the  sum  of'  their  squares  1 30,  and  the  sum  of  their  cities 
^80.  Ans.  6,   9,  2,  and  3. 

Let  w,  X,  y,  and  s  represent  the  numbers 


132  ROBINSON^S   SEQUEL. 

Then  because  they  are  in  proportion, 

wz=^xy  ( I ) 

By  conditions,     ?tf-|-a;-)-y+2;=20=a.  (2) 

And  w^+x^-^-y^'+z^^^lZO,  (3) 

And  ^3_(_^3^y3_|.23^98o.  (4) 

From  (2)  we  have      w-^-z^a — {x-\-y)  (5) 

w^-^^xoz+z^  =a^—2a{x-\-y)-{-x^-]-2xy-^y^ 
Suppressing  2wz  in  the  first  member,  and  its  equal  2xy  in  the 
second  member,  and  adding  (x^-\-y^)  to  both  members,  we  have 

w^^x^^y''-\-z''=a''—2a(x+y)-\-2x^-\-2y-' 
That  is  130=400— 2a(a:+y)+2a:2+2j^2 

Whence,  a{x+y)  =  l35-\-x^+y^  (6) 

By  cubing  (5),  we  have 

w^-\-32vz(w-^z)+z^=za^—3a^(x-\-y)-\'3a(x'\'yy 
—{x^-\-Sxy{x-\-y)-\-y^) 
By  transposing,  and  observing  that  Swz  equal  3xy,  we  have 
(w^-\-x^-\-y^-\-z^)-\~3xy{w-^x-\-y-\-z)=a^ — 3a^(a?+y) 
^        -^Sa(x-{-yy 
That  is,         9Q0-{-Saxy=za^—3a''(x-\-y)-j-3a(x-\-yy 
Dividing  by  20,  or  by  a,  which  is  the  same  thing,  and  we  have 

49+3x7j=a'—Sa{x+7/)-\.3{x+yy 
Or,  3xy=35\—3a{x-^y)-{-3{x+yy 

Dividing  by  3,  and  expanding  the  last  term,  gives 

xy=z  1 1 7 — a{x-\-7/)  ^x^ -{-2xy-\-y^ 
Or,  a{x+y)  =  \\7-^x--\-xy-{^/  (7) 

By  comparing  (6)  and  (7),  we  perceive  that 

iry+1 17=135 
Or,  xy=lS  (8) 

By  the  aid  of  (8),  (6)  becomes 

«2+2a:y+y2-|_135=20(ar+y)+3d 
Or,  (x-{-7jy—20(x+y)  =  —99' 

Whence,  (ar+y)— 10=±1 

Therefore,  x-\-y=\l,  or  9 


ALGEBRA.  13»' 

From  this  last  equation  and  equation  (8),  we  find  x=9,  or  6, 
and  y=2  or  3. 

(4.)     The  sum  of  Jive  numbers  in  geometrical  progression  is  31, 
and  the  sum  of  their  squares,  341.      What  are  the  numbers? 

Am.  1,  2,  4,     8,  and  16. 
Let  a:,  xy,  xy^,  xy^^  xy*  represent  the  numbers.         ^  ; 

Put         a=31  lla=341 

Then  x-\-xy-\-xy^  -\-^y  ^  -\-^y  ^  =^  ( 1 ) 

And      a;2-|-ar2y2_|.a2^4_|_^2y6_|_^2y8_iia  ^2) 

By  the  formula  for  the  sum  of  a  geometrical  series,  we  have 


-X 


y-1 


(3) 


And  ^''^'°~— =11«.  (4) 

Dividing  (4)  by  the  square  of  (3),  gives 

/ylO— 1\      /    y_l    y_ll 

Factoring  the  first  fraction, 

(y^+i)(y^-i)(y-i)(y-i)_n 
(y+i)(y-i)(y'-i)(y'— 1)    « 

Suppressing  common  factors. 


Divide  the  first  fraction,  numerator  and  denominator,  by  (y-f-1) 
and  the  second  fraction,  numerator  and  denominator,  by  {y — 1), 
and  we  then  have 

j^'+y^+y'+y+i    31 

Clearing  of  fractions  and  reducing, 

Here  we  observe  the  same  coefficients  whether  we  begin  to  the 
right  or  the  left  of  the  expression.  In  such  cases,  divide  by  half  the 
power  of  the  unknown  quantity.     In  this  example,  divide  by  y^ , 


■«v.l««^*i.'.,.j»«» 


1^  ROBINSON'S  SEQUEL. 

Then         lOy^— 2I2/+IO—  il+i^=0  (6) 

y    y"" 

Now  put        21y+-=P  (6) 

y 

Then  2/+^=^ 

y     21 

Squaring,  y2^2+l=^ 

10_  10P» 
2^       441 
Comparing  (5),  (6),  ?tnd  (7), we  perceive  that 

441 
10P2— 441P=4410 

F^—^-=b.     By  putting  6=441, 

Tlien  P^-lp  4-il=1^4^= W±00>. 

10       '  400     400  '  400 


10y^+20+^=J^_  (7) 


By  evolution,       F — — =J 

''  91)      M 


441-84}        .  21-29 


20      ^l      400  20 

p__21 -21+21  •29__21  •  60_105 
20  20'      ~Y' 

Or,  P=II?L.^=_1^ 

20  5 

Now  from  (6),  y4--=->  <^r  — - 

w2__%=^__i.     y^2,  or  1 
2  .  2 

To  find  a?,  we  must  return  to  equation  (3),  and  in  place  of  y, 

put  in  its  value  2,  and  x=\. 

Hence,  the  numbers  are  1,  2,  4,   8,  and  16. 

(5.)  There  are  six  numbers  in  geometrical  progression  ;  the  sum 
of  the  extremes  is  99,  and  the  sum  of  the  four  means  is  90.  Whai 
are  the  numbers  .^  Ans.  3,  6,   12,  24,  48,  96. 


Let  X,  xg,  xy^  ^  &c.  represent  the  numbers. 


ALGEBRA.  136 

Then  xi/^+x=99  (1) 

And  ary^  -^-xy^+xy^  -[-a:y=90  (2) 

Dividing  (1)  by  (2),  gives 

Dividing  numerator  and  denominator  by  (y+l)»  then 

'y^+y  10 

Clearing  of  fractions, 

10y4_iOy3_|_iOy2_iOy_(-io=lly3_|_iiy 

Or,  10y4— 21y3_|-.l03/2_2iy_|_io=0 

This  is  the  same  equation  as  (5),  in  the  preceding  example ; 
therefore,  as  in  that  example,  y=2. 

Now  from  equation  (1),  we  have  33a:=99,  or  a; =3,  the  first 
number. 

(6.)  The  number  of  deaths  in  a  besieged  garrison  atnounted  to  6 
daily;  and  allowing  for  this  diminution,  their  stock  of  provisions 
was  sufficient  to  last  8  days.  But  on  the  evening  of  the  sixth  day 
100  men  were  killed  in  a  sally,  and  afterwards,  the  mortality  increas- 
ed to  10  daily.  Supposing  their  stock  of  provisions  unconsumed  at 
the  end  of  the  sixth  day,  to  support  6  men  for  61  days  ;  it  is  requir- 
ed to  find  haw  long  it  would  support  the  garrison,  and  the  number  of 
men  alive  when  the  provisions  were  exhausted. 

Ans.  6  days,  and  26   men  aUve  when  the  provisions  were 
exhausted. 

Let  x^=  the  number  of  men  at  first, 

and  ji?=  the  amount  of  provisions  consumed  by  each  per  day. 

By  the  question,  we  have  a  decreasing  arithmetical  series, 
whose  common  difference  is  6*,  and  number  of  terms  8. 

For  the  amount  of  provisions  at  first  in  store,  we  have  px  for 
the  first  term,  and  {px — 42ji9)  for  the  last  term.  Then  the  sum 
of  the  terms  must  be  {^px — 168p). 

Hence,     8pa; — 168p=  the  amount  of  provisions  at  first. 
^px —  90p=:  the  provisions  consumed  in  6  days. 


Whence, 


2pa; —  78p=6jo-61,  the  provisions  left. 


136  ROBINSON'S  SEQUEL. 

During  the  6  days,  36  men  died,  and  100  men  were  killed ; 
therefore,  at  the  end  of  the  sixth  day  but  86  men  were  alive. 
Now  the  mortality  increased  to  10  daily,  and  at  the  end  of  n 
days  their  provisions  were  exhausted. 

Here  we  have  another  decreasing  arithmetical  series,  the  first 
term  86,  the  common  difference  10,  and  the  number  of  terms  n. 
The  last  term  is,  therefore, 

86— 10(7J— 1) 

First  term  86 

Sum  of  the  terms         (2-86— lOw+10)?? 

This  number  of  men  would  require 

(Q6n — 5n^-]-5n)p  amount  of  provisions. 

Therefore,  (91nr-5n'')p=6p'61 

Whence,  Sw^— 91w=— 366 

100»2—^+(91)2=(91)2— 366-20=8281— 7320=961 
By  evolution,  10/1—91=  ±31 

Whence,  w=6,  or  12^;  but  the  last  number  cannot  apply  to 
the  problem. 

(7.)    Out  of  a  vessel  cmdaining  24  gallons  of  pure  mne,  a  vintner 

drew  off  at  three  successive  times  a  certain  number  of  gallons,  which 

farmed  an  increasing  arithmetical  progression,  in  which  the  diff'er- 

en,ce  hetwem  the  squares  of  the  extremes  was  equal  ^o  16  times  the 

•mean,  and  filled  up  the  vessel  with  water  after  each  draught,  till  he 

fmind  what  he  last  drew  off,   reduced  to  one-sixth  of  its  original 

strength.     Required  the  number  of  gallons  of  pure  udne  drawn  off 

each  time. 

Ans.  12,  8,  and  3^. 

Let  X — y=  the  number  of  gallons  first  drawn. 

X      =  the  number  at  second  drawing. 
and  a;+y=  the  number  at  the  third  drawing. 
By  the  first  given  condition 

Axy==\Qx.        y=4. 
(a? — y),  or  (x — 4)=  the  pure  wine  at  the  first  drawing. 
He  then  filled  the  cask  with  water,  making  24  gallons  of  liquid, 
which  .contained  (24— a;+4),  or  (28— a;)  gallons  of  wine. 


ALGEBRA.  137 

Liquid.        Wine.        Liquid. 

^28— -a;  W 
Now  by  proportion,     24     :     28 — x     :    :    ar  :  ^^ 1—=.  the 

wine  taken  out  at  the  second  drawing. 

Therefore,  ("28 — x) — i — ZZz)—z=  the  wine  left  after  the  second 
^  ^  24 

drawing,  which  is  i Li — ZI_Z,  by  reduction. 

For  the  wine  taken  out  at  the  third  drawing,  we  have  the  fol- 
lowing proportion  ; 

24     •     (24-a;)  (28-a;)     .    .     ^  ,  4     .     (24-a:)(28--a^)(a:+4) 
24  '  24-24 

The  number  of  gallons  of  liquid   at  the  third  drawing  was 
(ar-[-4j^  but  only  one-sixth  of  it  was  wine  ;  therefore, 
(24— a;)  (28— a;)  (a;-f-4)_(a;+4) 
24-24  6 

Whence,  (jj^)  (24-.+4)^^ 

4-24 
Put  (24— a;)=P;  then 

P(P-|-4)=96 

F+SL^  ±10 
P=8  or— 12. 
That  is,  24 — x=^^,  or  x=\Q ;  the  minus  sign  is  not  appUcable. 
Hence  the  wine  at  the  first  drawing  was  x — 4,  or  12  ;  at  the  2d. 
(28— a;)a;    ^^  g     ^^^^  ^^  ^^^  2^  ^+4    ^^  ^^   ^^^  answer.. 
24  6  ' 

(8.)  There  are  four  numbers  in  geometrical  progression  such  that 
the  sum  of  the  extremes  is  56,  and  the  sum  of  the  means  24.  What 
are  the  numbers  r'  Ans.  2,  6,   18,  64. 

Let  X,  xy,  xy^ ,  and  xy^  represent  the  numbers. 
Then  by  the  conditions, 

a;y_|_a;2/2  _24=3a   .  (1) 

X  J^xy^=5Q=:la  {%) 

Dividing  (1)  by  (2),  and 

y(i+y)-3 
i+y3     7 


\ 


i 


13a  ROBINSON'S  SEQUEL. 

Dividing  numerator  and  denominator  by  (1+y),  and  we  have 

y       _3 


1— y+y^    7 

Or,  3y2_i0y-|-3=0 

If  y  is  a  whole  number  it  is  a  factor  of  3  ;  that  is  1  or  3.     It  is 
obviously  not  1 .     Try  3  ;  then 

3y-- 10+1=0,  ory=3. 
This  value  put  in  (1),  gives  \2x=24.     x=2. 

Another  Solution. 

Let  X,  and  y  be  the  means,  and  ~,  and  ?L  the  extremes. 

y  X 

Then  ^+?^=7a  (1) 

y      X 

And  a:+y=3a  (2) 

From  (1)  a;3-f.y3_7(^^y  ^3^ 

(2)  cubed,  x^+y^-\-Sxy(x+y)=27a^  (4) 

That  is,  7axy-\-9axy=27a^ 

16xy=:27a^  (6) 

Square  of  (2)  gives     x^ -\-2xy-\-y^  =:  9a^ 

27a2 
4xy= — -- 


By  subtraction,  x^—2xy+y''=  rr_ 

4 
a;— y=-|a  (6) 

From  (2)  and  (6),  we  readily  find  x  and  y. 


SECTION   VI. 

SOLUTIONS  OF  EQUATIONS  OF  THE  HIGHER  DEGREES. 

We  shall  take  equations  and  solve  them.  The  most  difficult 
equations  in  the  common  popular  books  will  be  selected,  begin- 
ning with 

newton's  method  of  approximation. 

(1.)     Given  x^-\-2x^ — 23ar=70,  to  find  one  value  of  x. 


ALGEBRA.  ia9 

By  trial  we  find  that  one  value  of  x  is  between  5  and  6,  nearer 
6  than  6 ;  therefore,  let  a=5  and  y=:  the  remaining  part  of  the 
root.     Then  a:=a-|-y. 

Expand,  neglecting  all  the  terms-  containing  the  powers  of  y 
after  the  first,  and  we  shall  have 

a:3=:  a3-|-3a2y-|-&c. 
2a;2=2a2-j-4ay  -\-kQ. 
—23x  =— 23a— 23y 
By  addition, 

In  this  last  equation  we  observe  Ihat  a  has  the  same  powers  and 
coefficients  as  x,  and  the  coefficients  to  y  may  be  found  by  the 
following 

Rule.  Multij)ly  each  coefficient  of  x  by  its  exponent,  diminish 
each  exponent  hy  unity,  and  change  x  to  a. 

Now  y= — it — .^       ?  ~~— .     Givinof  a  its  value,  5,  we  have 
^  3a^^4a—23  ° 

y=||  =  .l-[-.     Now  make  a=5.1,  and  substitute  again  in  the 

preceding  formula,  we  have  a  new  value  of  y. 

2*629 

Thus  y= =  .03.      Now  make   a=5. 13,  and  substitute 

^     75-43 

again,  aad  our  new  value  of  y  will  be  .0045784--     Hence,  a-j-y 

ora.=:5.134578+. 

(2.)     Given  a;4—3a;2-|-75ar=  10000,  to  find  one  value  of  x. 
By  trial  we  find  x  must  be  near  ten.     Hence,  put  a=10  and 
x=a-^y.     Then  by  the  preceding  rule 

Now  make  a=10 — .11=9.89.  If  we  have  the  patience  to 
substitute  this  value  for  a  in  the  equation,  we  shall  have  a  new 
value  to  y,  true  to  6  or  7  places  of  decimals,  and  of  course  a 
value  to  X  to  the  same  degree  of  exactness.  ^ 

(3.)  Given  3a;*— 35a;3— 1  la;^— 14a;+30=0,  to  find  one  value 
of  X. 

By  trial  we  find  that  x  mu;st  be  near  12.  Let  a=12,  and 
x:=a-\-y.     Then  by  the  rule  * 

\ 


140  ROBINSON'S  SEQUEL. 

^            12a3— 106a2_22a— 14  5338 

Hence, a;=12— .00112=11.99888. 


(4.)     Given  5x^—3x^—2x=\560,  to  find  x. 
We  find  by  trial  that  one  value  of  x  is  more  tlian   7.      Put 
«r=a-|-y,  and  a=7.     Then  by  the  rule 

156Q+2a+3a--5a3_  ^_  ^  ^^^ 

15a^—Sa—2            689  ' 

Hence, ar=7.00867+ 

We  give  Newton's  method  on  account  of  its  simplicity  in  prin- 
ciple; it  is  easily  understood,  and  can  long  be  retained  ;  but  its 
numerical  application  is  laborious  and  tedious. 

A  more  modern,  delicate,  scientific,  and  practical  method  is 
Homer's,  of  Bath,  England,  first  given  to  the  world  in  1819.  The 
principle  is  that  of  transforming  one  equation  into  another  whose 
roots  shall  be  less  in  value  by  a  given  quantity,  and  again  trans- 
forming that  equation  into  another  whose  roots  maybe  still  less,  &c. 
The  theory  is  fully  explained  in  Robinson's  Algebra,  last  two 
chapters. 

We  give  the  following  examples,  commencing  with  quadratics. 
Some  of  the  equations  here  solved  are  in  the  author's  class  book, 
and  are  numbered  as  in  that  work,  pages  313 — 333. 

(6.)  Given  a;2-|-7a:=l  194.  to  find  the  values  of  x  by  Homer's 
method. 

We  must  first  find  an  approximate  value  of  x  by  trial ;  but  the 
inexperienced  might  be  at  a  loss  how  to  make  the  trial.  We 
suggest  this  method  ;  separate  the  first  member  into  factors  thus  : 

Here  two  factors  of  1194  diflFer  by  7.  If  the  factors  were  equal, 
each  one  would  be  the  square  root  of  1194. 

Now  one  of  the  fac  tors  is  a  little  less  than  the  square  root  of 
1194,  and  the  other  a  little  greater  ;  but  we  want  the  less  factor. 

In  short,  the  square  root  of  1194  is  a  liUte  below  the  arithmetical 
mean  between  x  and  (x-\-l). 


ALGEBRA. 


141 


This  principle  will  do  for  a  guide,  when  the  coefficient  of  x  is 
small  in  relation  to  the  absolute  term.  The  square  root  of  1194 
is  34*5 ;  from  this  we  will  subtract  the  half  of  7,  3.5,  and  the  ap- 
proximate value  of  X  will  be  31  ;  therefore,  r=31.     a=7. 

T    si 


r+s 

38 
31.2 

1194(31.231 
1178 

1600 

a+2r+5 

s+t 

69.2 
23 

1384 
21600 

a+2r+2s+^ 

69.43 
31 
69461 

20829 
77100 

The  operation  may 
thus 

now  be 
69461) 

carried  on  as  in  simple  division  ; 

77100(111 

69461                                      ,i^ 

76390 

69461 

69290 
and  the  figures  thus  obtained  annexed  to  the  portion  of  the  root 
already  obtained.     Hence,  a:=31. 231111. 

As  the  sum  of  the  two  roots  is  equal  to  — 7,  the  other  root  will 
be— 38.231111. 


(7.)  Given  ic^— -2 l;r=2 1459 1760730,  to  find  theA^alues  of  x. 
Conceive  — 21a;  not  to  exist ;  then  the  value  of  x  will  be  the 
square  root  of  the  absolute  term  ;  but  this  term  has  six  periods 
of  two  figures  each,  and  the  superior  period  is  21,  the  greatest 
square  in  this  is  16,  root  4  ;  hence,  x  must  be  over  400000  ;  take 
r=400000.     a=— 21. 

-^+r  =399979 

r+5  460000 


— a-|-2r-f-5 

s-{-t 


214591760730(400000=r 
1599916      60000=5 


859979 
63000 


5460016 
5159874 


3000=  t 
200=u 


14S 


ROBINSON'S  SEQUEL. 


-^+2r+25+i: 

922979 
3200 

3001420 
2768937 

60±=t) 
l=w 

926179 
250 

2324837 
1852368 

926429 
51 

4724793 
4632145 

ar=46325L 

926480  926480 

926480 
A.S  the  algebraic  sum  of  the  two  roots  must  make  21,  the  other 
i-oot  must  be  —463230. 

We  can  find  the  negative  root  directly  as  well  as  indirectly,  by 
taking  r  minus  ;  then  s,  t,  u,  v,  &c.,  will  be  minus.  The  divisors 
and  quotients  both  being  minus,  their  products  will  be  plus. 

The  following  example  is  in  direct  contrast  to  the  preceding. 

{a)     Given  x^ — 32141a;=131,  to  find  the  values  of  x. 

Here  the  coefficient  of  the  first  power  of  the  unknown 'quantity 
is  large,  and  the  absolute  term  comparatively  very  small.  The 
factors  X  and  {x — 32141)  are  so  very  unequal,  that  a  resort  to  the 
square  root  of  the  absolute  term  for  an  approximate  value  of  x,  as 
in  the  preceding  equation,  would  be  useless.  In  this,  and  in  simi- 
lar cases,  we  can  obtain  an  approximate  value,  by  conceiving  the 
absolute  term  to  diminish  to  zero.     Then 

«2-^32141a;=0. 

This  equation  will  be  verified  by  putting  a;=0,  and  ar=32141  ; 
and  from  this  consideration  we  conclude  that  one  value  of  x  in 
our  equation  must  be  very  small,  and  the  other,  over  30000. 
Hence,  put  riti30000,  and  the  solution  is  as  follows. 

131(30000=r 


-.^ 

—2141 

—64230000 

r-H 

32000 

+64230131 

2000=* 

— a+2/--[-5 

29859 

59718000 

.+/ 

2100 

4512131 

100=< 

— <i+2r+2s-l-if 

31959 

3195900 

40=?« 

t^^u 

140 

1316231 

1=V 

ALGEBHA. 

32099 

1316231 
1283960 

41 

32271 

32140 

32140 

u» 


131 
Here  we  obserye  that  the  last  divisor  is  numerically  the  same 

as  the  coefficient  to  x,  and  the  last  dividend  is  131,  the  same  as 

the  absolute  term. 

Now  if  we  divide  131  by  32140,  we  shall  obtain  decimal  places 

in  the  root,  and  the  positive  value  of  x  will  be  32141  s^jIj,  very 

nearly. 

The  first  four  or  five  decimal  places  will  be  exactly :  then  the 

figures  will  be  too  large,  because  the  divisor  accurately  corrected, 

will  increase  a  little  at  every  step.     From  this  example,  we  learn 

that  when  we  have  an  equation  in  the  form 

x^ — ax=  zhb, 
and  a  numerically  greater  than  b,  the  positive  value  of  x  will  be 

(  «rt-  ),  very  nearly,  the  value  being  a  very  little  in  excess  of  the 

true  value,  and  if  -  is  a  small  fraction,  this  approximate  value 

of  X  will  be  sufficiently  near  to  call  it  the  true  value. 
When  the  equation  is  in  the  form 

x^  +««=  dr&, 
and  a  greater  than  b,  then  the  negative  value  of  x  will  be  express- 
ed by  — (  adz-  j,  very  nearly,  and  if  b  is  much  greater  than  d, 
we  may  say  accurately,  in  a  practical  point  of  view. 

If  we  take  the  equation  x^ — ax=^b,  and  take  a;=a-|--,  and  at- 

a 

tempt  to  verify  the  equation  with  this  value,  we  shall  have 

a2+26+^— a^— 6=5,     Or,  —=0 

a^  a^  ^ 

The  error  then,  is  — ;  and  thus  we  perceive  that  if  -  is  a  small 


^44  ROBINSON'S  SEQUEL. 

fraction,  the  approximate  value  of  x  is  really  found ;  but  if  h 
is  greater  than  a,  it  can  hardly  be  called  an  approximation, 

EXAMPLES. 

{b)  Given  x^ — 3165a:=632,  to  find  the  approidmate  values 
of  X.  Ans.  x=3165^\%%,  or  — gVeV 

(c)  Given  x' — 2178:ir= — 69,  to  find  the  approximate  values 
of  X,  Ans.  a;=2178— 2^8^  or  2ff t- 

(d)  Given  a;2-j-3116a;=141,  to  find  the  approximate  values 
of  X,  Ans.  x=  -(31 I6+3VV6 ),  or  3VVe , 

(e)  Given  a;^-|-591a;= — 71,  to  find  the  approximate  values 
of  X,  Am.  x= — 591-|-5Vi,  or — /Jy- 

The  foregoing  values  are  so  near  the  true  values,  that  they 
would  be  taken  for  true  values,  in  any  practical  application. 

(/)     Given  x^-\-'^^x= — 4,  to  find  the  values  of  x. 

Ans.  x= — y-(-^= — 8  nearly. 
— 8  is  the^  exact  negative  value  of  x,  and  |-  is  the  positive  val- 
ue.    We  get  y\  for  the  approximate  positive  value. 

(ff)     Given  x^ — ^x= — 1,  to  find  the  approximate  values  of  ar. 
Am.  x=^ — |=T^  >  the  true  value  is  2. 

(A)     Given  a;^ — %^^= — 1,  to  find  the  approximate  values  of  x. 
Ans.  a:=2j^ — 2^6=^  nearly  ;  5  is  the  true  value. 
The  other  value  is  ^. 

When  a  and  b  are  equal,  or  nearly  equal,  in  the  equation 
x^dzax=±.b, 
I  it  is  most  difficult  to  find  the  value  of  r,  or  the  approximate  val- 
ue of  x\ 

Having  now  sufficiently  explained  the  means  of  finding  r  in  the 
different  cases,  we  resume  the  appUcation  of  Horner's  method  of 
operation. 

(8.)     Given   Ix^ — 3a;=376,  to  find  one  value  of  ar. 


ALGEBRA.  145 

Or  x^—^x=^^.     TxLtx==}y.     (Art.  166.) 

Then    ^—^=?IA.     Or,  y2—3y=376X 7=2625. 
49     49       7 

In  this  equation  we  perceive  that  y  must  be  more  than  the 

square  root  of  2625,  that  is,  more  than  50.     Hence,  put  r=50. 

rs   t 


T+S 

47 
52 

2625(62.766+ 
235 

99 

2.7 

275 
198 

1017 

7700 

75 

7119 

10246 

68100 

51225 

6875 

Hence  x= — '- Zt=7.+ 

7  ^ 


(10.)     Given  x^ — y3ya;=8,  to  find  x  true  to  seven  places  of 
decimals. 

Put  a;=-^ ;  then  a;^=-^-^,  and  the  given  equation  is  trans- 
11  121  ^  ^ 

formed  into  ^    — _^=8. 
121      121 

Or,  y2_3y_9g8.  .,-.% 

It  is  obvious  that  y  must  be  between  30  and  40 ;  therefore, 
r=30. 


-«+r 

27 
32 

968  (  30. 
810 

_-a+2r+s 

s+t 

b9 
2.6 

158(2.64883626 
118 

61.6 
64 

4000 
3696 

6224 
48 

30400 
24896 

10 


146 


ROBINSON'S  SEQUEL. 


62288 

660400 

88 

498304 

62|2|9|6|8 

5209600 

4983744 

226856 

186890 

38966 

37376 

1590 

1245 

346 


After  the  4th  decimal,  the  operation  was  carried  on  by  con- 
tracted division,  giving  ^=32.64883625. 

But  x=  ^r  of  y  ;  therefore,  ic=2.96807602. 


Put    ar=i- ;  then  «*  =-t— — ,  and  the  equation  becomes 
^R  (36)2  ^ 


(11.)     Given  4a;^-j-^ar=i,  to  find  one  value  of  x. 
This  equation  is  the  same  as  x^-\-:J^x=^-^. 

;  thena;*=J''' 
36 

(36)2~(36)2     20 

Or,  y^+7y=G4.8 

It  is  obvious  that  the  value  of  y  is  between  6  and  6 ;  therefore 
r=s6.     a=7. 


«+r 

12 

64.8  (  5.277812946 

r+s 

5.2 

60 

a+2r+s 

17.2 

480- 

8+t 

2.7 

544 

17.47 

13600 

77 

12229 

17647 

137100 

78 

122829 

176648 

1427100 
1404384 

ALGEBRA. 

227160 
175548 


147 


17|6|614 


51612 
35108 
16504 
15788 


816 

702 

114 
Whence,  ^=6.277812*946,  or  —12.277812946, 
And  a;=0. 146605915,  or  —0,341050359, 


(12.)     Given   iX^-]--5X=^j,  to  find  one  value  of  x. 

Put  x=l 
5 

4y_28 


2  ,  4a:     28 
'    5      33 


Then 


r_+ 

25  '  25     33 
Or,  5^2_j_4y_7_oj)^2l,21212121 

r=:3.     a=4. 


u+r 

7. 

21.21212121, 

&c,  (3.021186235, 

r+s 

3 

21 

a+2r+8 

10.0 

2121 

.+t 

02 

2002 

1002 

11921 

21 

10041 

10041 

188021 

11 

100421 

100421 

8760021 

18 

8033824 

10042|2|8 

626197 
602536 

23661 
20084 

3577 


148  ROBINSON'S  SEQUEL. 

y=3.021 186235.     x =0.60  4237  257. 


(13.)     Given    116 — 3x^ — 7.c=0,  to  find  one  value  of  x. 
^3        3  3 


Then.  3^+!3?=ll^ 

9       9        3 

y2_j_7y=:345.         r=16.         a=7. 

a+r  22  345(15.40158. 

r-{s  15.4  330 

a-{-2r-^-s  37  A  1500 

s+t  40        1496 

aJ^2r+2s-\-t  37|801         40000 

37801 

2199 
1890 

309 

302 

^=15.40158.  ar=5.13386. 

We  will  now  apply  Horner's  method  to  cubic  and  the  higher 
equations.     For  the  theory,  we  must  go  to  the  class  books. 

CUBIC    EQUATIONS. 

The  first  example  here,  is  the  third  on  pag,e  319  of  the  author's 
class  book.     Hence, 

(3.)     Given  x^^2x^ — 23a;=70,  to  find  one  value  of  x. 

By  trial  we  find  x  must  be  a  little  over  5 ;  therefore, 

r=5,  A=2,  JB=— 23,  iV=70 

B —23 

r(r+A) 35^  r  8t 

1st  Divisor 12  ^  70  (  6.134 

r» 26j  60 


ALGEBRA.  "lit 

B' 72  10000 

5(5+3r+^) ATi\  7371 

2d  Divisor 7371  \  2629000 

»2 1  J       2276697 

B" 7543         352303000 

*(^Q-\-t)t ...4599       305649104 

3d  Divisor 758899  46653896 

e 9 

B'" 763507 

61576 

4tli  Divisor 76412276 

16^ 

Common  division  will  give  three  or  four  more  figures  to  perfect 
accuracy. 

(4.)     Given  x^ — 17a;^-|-42a;=185,  to  find  one  value  of  x. 

Here  ^=—17,  ^=42,  iV^=185,  and  we  find  by  trial  that  x 
must  be  between  15  and  16  ;  therefore,  r=15. 

B 42 

r{r+A) —30]  r  st 

1st  Divisor ~12  ^  185  (  15.02 

H 225  J  •       18Q      

B' 207  5000000 

*(s+37-+^) __0]  4154008 

2d  Divisor 207  \  2Q11)  "845992  (407 

«2 oj  8298 

J5" 207  16192 

t{^Q+t) 7004  14539 

3d  Divisor 2077004  1653.0 

Hence, ir=15.02407+ 

(5.)     Given  x^-\'X^  =500,  to  find  one  value  of  x. 
Here  A=\,  B=0,  r=7. 

*Q  represents  the  root  as  far  as  previously  determined 


tSO  ROBINSON'S  SEQUEL. 

B 0 

r(r~{-A) 56^ 

1st  Divisior 56  [  600  (7.61 

r« 49J  '       392 

^ 161  108 

(Sr-\-s-\'A)s 1356  104736 

2(1  Divisor 17466  3264 

8' 36  1887181 

18848  )  1376819 
2381 

Sd  Divisor 1887181     Continue  by  common  division. 

1 

1889563 

(6.)     Given  x^-\-10x^-{-5x=2600,  to  find  one  value  of  ar. 
Here   ^=10,  B=5,  r=ll. 

B 5  2600  (  U  .  006 

r(r+A) ^3J  ^  2596 

1st  Divisor 236  \  4 

r^ j  2^  J  3529188216 

B' 588  470811784 

(3E-\-u)u 198036  Continue  by  common 

4th  Divisor .588198036  division. 

(6.)     Find  one  value  of  a:  from   5x^ — 6x^-{-3x= — 85. 

As  the  result  is  negative,  we  will  change  the  second  and  every 
alternate  sign  of  the  equation,  (Art.  178),  and  find  a  value  of  x 
from  the  equation         5x^ -\-6x^ -\-3x=85. 

Use  the  formula  of  (Art.   194).     c=5,  A=6,  -5=3,  and  by 

trial  we  find  r=2. 

r  s 

B 3  85(2.1 

{cr+A)r..,. .32^  70 

1st  Divisor 35  f  16 

or^ 20  J  9.066 

"87  5.935 

{^+C8+A)s ,_ZSb  Continuing  this,  we  shall  find 

2d  Divisor 90.65  the  value  of  x  to  be  2.1 6399+, 

^*   5  and  its  sign  changed  will  be  the 

94.35  value  of  x  in  the  original  equa. 


ALGEBRA.  '161 

(7.)     Find  x  from  the  equation  \2x^-\-x^ — 6a;=330. 
Here  c=12,  ^=1,  jB=— 6,  r=3. 

B ....—6 

(cr+A)r .ml  »•  «^ 

1st  Divisor 106  f  330(3.036 

cr^ 108  J  31£ 

5'. 326  ir  , 

(3ci2+cO^ 11208  9783624  \, 

3d  Divisor 3261208  2216376 

c^ 108  Continue  thus. 

3272624 
In  the  same  manner  perform  8  and  9. 

(Art.  196.)  Page  323. 

(3.)     Extract  the  cube  root  of  l-352-606-460'694'688 
For  the  sake  of  brevity,  take  r=ll,  in  place  of  1. 

1st  Divisor 121  rst' 

B'=^r^ 363  1-352-606-460-694-688  (  110692. 

{pB+t)t 16525     1331 

2d  Divisor. . .  .3646526       21  605  460 
25        18  232  625 

3663075  3  372  835  594 

{ZB-\-u)u,,:.        298431  3  299  453  379 
3d  Divisor...: 366605931         73  382  215  688 
81         73  382  215  688 

366904443 

(3i2+2;> 663544 

36691107844 

(4.)     By  the  table  of  cubes  which  run  to  8000,  we  perceive 
at  once  that  r  in  this  example  is  17. 

1st  Divisor 289  r  s  t 

3r2  =B' 867  6382674  (  175.2        i 

{^RJ^s)s 2576  4913 

2d  Divisor 89276  469674 

26  446376 


IdS! 


ROBINSON'S  SEQUEL. 


91876 

(SB+ty 10604 

3d  Divisor 9198004 

4 


9208612 


23299000 

18396008 
4902992 
Complete  another  divisor,  then 
continue  as  in  simple  division. 


(6.)     Find  x  from  the  equation  a;3  =  16926.972604. 
For  the  sake  of  brevity,  let  r  represent  the  value  of  the  two 
superior  digits.     That  is,  let  r=26. 


Ist  Divisor.... 626 

£'=3r^ 1876 

(3r+s> 761 

2d  Divisor...  188261 

8^ 1 

B" 189003 

(SM+ty 46216 

3d  Divisor....  18946616 

36 

18990768 
7648 

4th  Divisor      1899084348 


16926.972604  (  26.16002649 
16626 


301  972 
188  261 

113  721604 
113  673096 


48408  000  000 
Common  division. 
189.91  )  48408  (  2649 


37982 

10426 

9496 


931 

769 
172 


It  is  not  important  to  show  a  solution  to  the  remaining  exam- 
ples under  this  article. 

From  Robinson's  Algebra,  page  324. 


(1.)     Given 

x^+x^+x^ 

—X 

=600,  to  find 

one  vah 

By  trial,  we 

find  r=4. 

1              1 

1 

—1             = 

=600  (  4. 

4 

20 

84 

332 

6 

21 

83 

168 

4 

se 

228 

. 

ALGEBRA. 

1          9 

67            311 

4 
13 

17 

62 
109 

A  new  transformed  equation  is 

5^+17* 

3^109s^+31l5=168. 

IBS 


'' 3  11  — ^ 

17.  109.  311.  =168.(0.4 

.4  6.96  46.384  142.9536 

17.4  115.96  357.384  25.0464 

4  7.12  49.232 


17.8    123.08    406.616 
4      7.28 


18.2    130.36 

4 

18.6 
The  next  transformed  equation  is 

t*     +18.6^3     _|_130.36^2     _|_406.616^     =25.0464(0.06 
.06  1.12  7.888  24.8602 

.1862 


18.66 
.06 

131.48 
1.12 

414.504 
7.956 

18.72 
.06 

132.60 
1.13 

422.460 

18.78 
.06 

133.73 

18.84 
Several  other  decimal  places  of  the  root  may  be  found  by  the 
following  division, — the  powers  of  u  above  the  first  being  consid- 
ered valueless, 

423.  )  0.1862  (  0.00044019 
1692 


i  1700 


1692 


800 
Hence  the  root  is  4.46044019. 


164  ROBINSON'S  SEQUEL. 

(2.)     Given  x^—5x^-}-0x''-^9x=2.Q,  to  find  one  value  of  x, 
r=0.3,  found  by  trial. 


—6. 

0.3 

—4.7 

+0. 

—1.41 

—1.41 

+9. 
—.423 

8.677 

r 
=2.8  (  0.3 
2.6731 

.2269 

.3 

—1.32 

—.819 

—4.4 

—2.73 

7-768 

.3 

—1.23 

—4.1 
.3 

—3.96 

—3.8 
54_3.8s3_3.96s2-|-7.768«=0.2269. 

-3.8  —3.96  +7.758     =0.2269(0.02 

.02  —.0766         —0.081         0.16364 


—3.78         —4.0366  7.677  .07336 

.02  .076  .082 


—3.76         —4.11  7.696 

The  remaining  figures  may  be  found  by  division,  thus  : 

7.6  )  .07336  (  0.00978 
676 


686 
625 


610 
Hence  the  approximate  value  of  x  is  0.32978. 

(3.)    Given  x* — 9x^ — 1  la:*— 20a;=  — 4,  to  find  one  value  of  x. 

r 


1 

—9. 
.1 

—11. 

—.89 

—20. 
—1.189 

=-4(.l 
—2.1189 

—8.9 
.1 

8.8 
.1 

—11.89 
—.88 

—12.77 
—.87 

—21.189 
—  1.277 

—1.8811 

—22.466 

—8.7 
.1 

—13.64 

1 

—8.6 

—13.64 

—22.466 

—1.8811  (.07  &c. 

ALGEBRA.  166 


-8.6  —13.64  —22.466       =—1.8811  ( .07 

.07  —.597  —.9966         —1.642382 


—8.63        —14.237         —23.4626         —  .238718 
.07  —.592  —1.0380 


—8.46         —14.829         —24.6006 
.07  —.587 


-8.39         —16.416 
.07 


-8.32 


1     —8.32         —15.416         —24.6       =—0.238718(0.009 
.009  —.0748  —.139       —  .2217647 


■8.311       —15.4908       —24.639       —  .0169633 
9  —.0747  —.140 


—8.302       —15M55      —24.779 
9  —.0746 


—8.293       —16.6401 
9 


-8.284 


—24.78  )— .0169633  (  0.900684 
14868 
20953 
19824 


11290 
9912 
13780 

Whence,    a;=0. 179684,  nearly. 

(4.)     Given  x^=z5000,  to  find  one  value  of  a?,  or  we  may  say 
find  the  fifth  root  of  5000. 

Here  all  the  coefficients  are  zero,  except  the  first,  and  r=5. 


1^  ROBINSON'S  SEQUEL. 


r 

1    0 

0 

0 

0   =6000  (  6 

6 

26 

125 

625     3126 

5 

26 

126 

625     1876 

5 

60 

376 

2600 

10 

76 

500 

3126 

6 

75 

750 

16 

150 

1250 

6 

100 

20 

250 

5 

26 

1   26. 

250. 

1250. 

3125.    =1875.  (  .4 

.4 

10.16 

104.06 

541.624   1466.6496 

26.4 

260.16 

1354.06 

3666.624    408.3504 

.4 

10.32 

108.19 

584.900 

25.8 

270.48 

1462.25 

4251.524 

.4 

10.48 

112.38 

26.2 

280.96 

1574.63 

.4 

10.64 

26.6 

291.60 

.4 

27.0 

1   27. 

291.6 

1674.6 

4251.52  =408.3509(.09 

.09 

22.38 

28.26 

144.26   395.6202 

27.09 

313.98 

1602.86 

4395.78    12.7307 

.09 

24.46 

30.46 

146.998 

27.18 

338.44 

1633.32 

4542.778 

Now  by  common  division, 


4642.7|78  )  12.7307 (  .0028,  nearly. 
9.0866 


3.64i52 
Whence,    a;= 5.4928,  nearly. 


ALGEBRA.  157 

It  is  practically  useless  to  solve  such  equations  as  the  preced- 
ing, because  solutions  are  so  simple  and  direct  by  logarithms. 

(5.)     Given   x^  =  (-^)  ' ,  or  a:^  = ^! ,  to  find  one 


value  of  X, 


+1/  x^-{-2x^  +  l 

a;5-|-2a;3+a;=64.         r=2 

0  2           0             1         =64  (  2. 

2  4  12           24             50 

2  6  12           25             14 

2  _8  28           80 

4  14  40         105 

2  12  52 

6  26  92 

2  16 

8  42 

_2 

10 

10.  42.               92.               105.         =14(0.1 

.1  1.01              4.3                 9.63         11.463 


10.1         43.01  96.3  114.63  2.537 

.1  1.02  4.403  10.07 


10.2 
.1 

10.3 
.1 

44.03 
1.03 

45.06 
1.04 

10.4 
.1 

46.1 

10.5 

10.5 

.02 

46.1 
.21 

10.52 
.02 

46.31 
.21 

100.703  124.70 

4.506 


105.209 


105.21  124.7       =2.537(0.02 

.926  2.1227     2.536454 


106.136  126.8227       .000546 

.93  2.141 


10.54         46.52         107.066  128.9637 

128.96  )  .00054600  (  0.000004+ 
51584 
Whence,    a?=2. 120004,  nearly. 
By  an  exact  solution,  the  last  figure  would  be  3>  in  place  of  4. 


158  ROBINSON'S  SEQUEL. 

Observation.  When  we  observe  that  the  sum  of  the  coeffi- 
cients in  any  equation  is  zero,  we  may  be  sure  that  unity  is  one 
root  of  the  equation. 

Then  we  can  depress  the  equation  one  degree  by  division.  For 
example,  we  are  sure  that  the  equation 

has  a  root  =1,  because    1 — 7-|-7 — 1=0  ;   and  1  put  for  x  will 
neither  increase  nor  decrease  any  of  the  terms. 

The  equation  x^-^-'ix^ — Ix^ — 8a?-}-12=0,  also  has  one  root 
=  1,  for  the  same  reason. 

lii  Bland's  problems  I  find  the  following  problem,  (page  426). 
One  root  of  the  equation  x'^ — hx^ — ^a;-|-6=0,  is  5;  determine  all 
the  roots.  Here  we  perceive  that  another  root  is  1  ;  therefore, 
the  equation  is  divisible  by  (x — 5)  (if — 1),  or  by  x^ — 6ar-|-6 ;  thus 

x^^QxJ^b  )  a;4— 5a;'— ar+5  (  aj^+ar-f-l 
a:4_6a;3_|_5a;2 

x^ — ^x^-\-bx 


x^—Qx-\-b 
a;2_6a?-f5     • 

Whence  a:2-|-a;+l=0,  and  x—  i(±V^^— 1)- 

To  solve  some  of  the  following  problems,  it  may  be  necessary 
to  see  how  the  roots  combine  to  form  the  coefl&cients. 

We  shall  consider  all  the  roots  as  positive;  and  represent  them 
by  a,  5,  c,  c?,  e,  &c. 

Then  an  equation  of  the  second  degree  will  be  represented  by 

a;2__^^_^j_^0.  (1) 

An  equation  of  the  third  degree,  by 


x^-^-a 
—5 


x^-\-ab 
4-ac 


x-^abc^O>  (2) 


--cb 
An  equation  of  the  fourth  degree,  by 


x^ 


ALGEBRA. 

rc-|-a5cc?=0. 


169 


—a 

;r3- 

-aS 

x^  —  abc 

—5 

-(m 

-^bd 

— c 

_ 

-ad 

— acd 

— rf 

- 

-cb 

-—cbd 

_ 

-cd 

H 

-bd 

(3) 


Now^  let    A=a+b-\-^+d.      £=:a(b+c+d)-[^(c+d)-\-cd. 

C=a(bc+bd-{-cd)-{-cbd.  J)=abcd. 

Then  the  equation  of  the  fourth  degree  will  become 

x*—Ax^-\-Bx^^Cx-]-D=0.  (4) 

This  equation  multipled  by  (x — U),  gives  the  representative 
of  an  equation  of  the  fifth  degree,  as  follows  : 

-£JD=0        (5) 


x^—A 
—E 


a;4+    B 
\-EA 


^3_     C\x^J^    J) 
—EB\     -\-ED 


In  these  equations  we  observe  that  the  coefficient  of  the  high- 
est power  of  a:  is  1  ;  and  that  the  coefficient  of  the  next  inferior 
power  is  the  sum  of  all  the  roots,  with  their  signs  changed ; — ^the 
absolute  term  is  the  product  of  all  the  roots. 


EXAMPLES. 

(1.)     The  roots  of  the  equation 

6a:*— 43a;3-j-107a;2— 108a;+36=0, 

are  of  the  form  a,b,~,  ~ ,  find  them. 
a     b 

Divide  the  equation  by  6,  so  that  it  may  compare  with  the 
fundamental  equations  (3)  or  (4),  then 

Now  if  the  roots  are  of  the  form  a,  5,  _,  -,  we  may  take  these 

a     b 

symbols  to  represent  the  roots. 
Then  a 


^+^!+^=13,  and  a6=6. 
^      ab  6 


That  is,  6(«+5)-l-a2+52  =43. 

By  adding   2a5=:12  to  the  last  equation,  we  have 


160  ROBINSON'S  SEQUEL. 

Whence,     a-\-b=5 ;  but  ab=6.     a=2.     b=3. 
And  the  roots  are  2,  3,  f,  |. 

(2.)     The  roots  of  the  equation 

a;4  _i0a;3 +352:2— 50a;+24=0, 
are  of  the  form  (a+1),  (a—1),  (6+1),  (5—1),  find  them. 
Here    2a+26=10.     (a^— 1)  (6^— 1)=24. 
That  is,  a+b=5.     a^S^—a^— 62+1=24. 
But  2a6+a2+62==25. 

By  addition,  a'b''-\-2ab-Jf-l=49. 

a6+l=7,  or  a6=6. 

Buta+6=5;  hence,  a=2,  6=3,  and  the  roots  are  1,  2,  3, 
and  4. 

The  roots  of  the  following  equations  are  in  arithmetical  pro- 
gression ;  find  them. 

1."        a;3_6a;2— 4ar+24=0. 

2.  x^—9x^+^3x—ie=0: 

3.  x^—'6x^+llx—6=0. 

4.  a;4— 8a;3+14a;2+8a;— 15=0. 

5.  a;^+a;3— lla:2+9ar+18=0. 

We  work  out  the  fifth  and  last  example  ;  it  being  the  only 
difficult  one. 

Let  (a— 36),  (a — 6),  (a+6),  and  (a+36),  represent  the  roots ; 
then  4a=— 1,  and  (a''—b^)(a^—9b^)=lS. 

That  is,     a^— 10a2&2_|_96^  =  18.     Buta2=-i.V,  «*  =  2-k- 

Therefore,  -1 1^+96^  =  18. 

266       16     ' 

_i___1062+14464=288. 
Or,  1446t— 1062=288— yV- 

Add  j''^^  to  both  members  to  complete  the  square. 

Then         1446^--1062+yV_=288yV4=^Hf^ 
By  evolution,         1262— y5_=2_o_3^fi|&oi 

Whence,  62=1.449209 


ALGEBRA.  161 

And  5=1.20383.         But  a=— 0.25. 

Therefore,     a— 35=— 3.86149,     log.       0.586761. 
a—  5=— 1.45383,     log.       0.162515. 
a+  5= +0.95383,     log.  —1.979458. 
a+35= +3.36149,     log.       0.526510. 
Log.  18,  1.255244,  nearly. 

The  sum  of  these  numbers  is  — 1,  as  it  ought  to  be,  and  the 
product  of  the  roots  is  18. 

Negative  numbers  have  no  logarithms,  because  there  are  in 
fact  no  such  numbers.  The  product  of  several  numbers  is 
numerically  the  same,  whether  the  numbers  be  positive  or  neg- 
ative ;  therefore,  we  took  the  logarithm  of  each  root  as  though 
it  were  positive.  The  product  in  every  case  will  be  positive  or 
negative,  according  as  the  number  of  minus  factors  is  even  or 
odd. 

The  roots  of  the  following  equations  are  in  geometrical  progres- 
sion :  find  them. 

1.  a;3— 7:r2+14;r— 8=0. 

2.  a;3— 13a;2_|_39_^_27^0. 

3.  a;3— 14ic2+56a:— 64=0. 

4.  a;3—26a:2+156a;— 216=0. 
Let  a,  ar,  and  ar^  represent  the  three  roots. 
Then     a-\-ar-\-ar^=2Qy  and  a^r^=:9,\Q. 

From  these  equations  we  find  a=2  and  r=3,  and  the  roots 
sought  are  2,  6,  and  18. 

Problems  like  the  preceding  are  impractical,  because  there  is 
no  natural  method  of  finding  the  form  of  roots,  a  priori,  and  to 
give  the  form,  is  nearly  equivalent  to  giving  the  roots  themselves. 


RECURRING  EQUATIONS. 


A  recnrrrng  equation  is  one  in  which  there  is  a  symmetry  among 
the  coefficients ; — the  terms  which  are  equally  distant  from  the 
extremes,  have  the  same  numerical  coefficient.     For  example, 
11 


162  ROBINSON'S  SEQUEL. 

is  a  recurring  equation,  for  the  coefficients  are   1,  — 11,  +17, 
which  recur  in  the  inverse  order  17,  — 11,  1. 

Here  it  is  obvious  that  x= — 1  will  satisfy  the  equation  ;  and 
if  we  change  the  second  and  every  alternate  sign,  then  x=  1  will- 
satisfy  the  equation  ;  that  is,  1  is  a  root  of  the  equation 

x^+nx'-\-17x^^l7x^—Ux—l=0. 

Here  the  sum  of  the  coefficients  is  0,  and  consequently  x=1 
must  satisfy  the  equation. 

Now  we  arrive,  at  this  general  truth  : 

A  recurring  equation  of  an  odd  decree  will  have  either  — 1  or  +1 
/or  one  of  its  roots. 

It  will  have  — 1,  if  the  corresponding  coefficients  have  like  signs — 
and  -|-1,  if  they  have  unlike  signs. 

Hence,  every  recurring  equation  of  an  odd  degree  will  be  di- 
visible by  (x-{-\)  or  (x — 1),  and  can  thus  be  reduced  to  an  equa- 
tion of  an  even  degree,  and  one  degree  lower  than  the  original 
equation. 

Every  binomial  eqtcation  is  also  a  recurring  equation. 

Every  recurring  equation  of  an  even  degree  above  the  second, 
can  be  depressed  one  half  by  the  following  artifice  : 

Take  the  equation, 

x^+5x^-\-2x^+5x+l=z0. 

Divide  every  term  by  the  square  root  of  the  highest  power,  in 
this  example  by  ic^  ;  then 

Then       •  (.»+^)+5(.+l)+2=0. 

Put    x+l=y;  thenx^+24-—=y\ 

X  x^ 

Whence,  y^-^oyz=0,  an  equation  of  only  half  the  degree 
of  the  given  equation. 

This  can  be  verified  by  y=0,  or  y= — 5.. 


ALGEBRA.  }6S 

Therefore,  ar-|-_=0,  or  x4-^  =  — 5. 

X  X 

Whence,        a?=±V— 1,  or  x=\{—b^J1\) 

Find  all  the  roots  of  the  equation  x'= — 1=0, 

One  is  obviously  one  root,  therefore  divide  by  x — 1=0. 

Then        x'-^x^-\-x''-\-x^\=^.     Divide  by  a:2. 

x     x^ 

Put     :c+l=y;  then    x^ +2+^^=1/^ 
X  x'^ 

Whence,  y^-\-y — 1=0. 

y=-i+iV5,  or  y=-i_i75.  ^ 

Put  2a=_i-(-i^5;  then  — (l+2a)=— ^— i^5. 

Now   x-\--,=2a,  and  x-{-  = — (l+2a) 

X  X 

From  the  first,     x=a-\-jJa^ — 1,   or  x=a-^Ja^ — 1 

That  is,  x=\{J^—l+J^^^^l0^2j5) 

Or,  ^=J(^5— 1— VZTc^-SVS) 

Taking  the  second. 

Or,  ^•=— KV^+i+V—i^+VsJ 

Given  (^'^+1)  (.^'-  +  l)  (^+l)=30.r^  to  find  the  values' of  x. 
Multiply  as  indicated,  the  product  is 

a;  B -(-a;5 +;c  ^ +22- 3 -[-a;2 -[-.r-{- 1  =30^-3 
Dividing  by  x^,  and 

X     x^     x^ 
0.  (.3+^.)+(.+^)V(.+l)=30 

Put       (-+;)=y.     Then  (.3+^)+3(.+»)=,» 


164  ROBINSON'S  SEQUEL. 

Whence,  y3_3^_|_y2_|_y_3Q 

Or,  y3_|_y2_2y^30. 

The  first  attempt  at  solving  this  by  Horner's  method  shows  us 
that  r  is  3  exactly  ;  that  is,  y=3. 

Then  ar+i  =3,  and  rr=i(3db75.) 

The  other  values  of  y  are  imaginary. 

Given    {x-\-y)  {xy-{-\)=^\^xy  (1) 

and    (a:2-j-y^)(a;2^2_(_ij_.2O0^2y2  ^g^ 

to  find  the  values  of  x  and  y. 

Multiply  as  indicated,  then 

x''y-\-xy^-^x^y=nxy  (3) 

a:4?/2_j_^2^4_j_^2_|_^2^2082;2y»  (4) 

Divide  (3)  by  xy,  and  (4)  by  x^y^ ,  then 

^+2/+-+-=  18  (5) 

y    X 
And  a;2-|-y2+_L-|-Jl.=208  (6) 

Now  put   P=^a;+1Y  and  q=(vJ^\. 

Then  P_[-§=18  (7) 

And  P2_|_^2^212'  (8) 

From  (7),  P2_|.2P^_j.^2^324  (9) 

2P^=112  (10) 

(10)  from  (8)  gives 

P~Q=±zlO 
Whence,  P=14  or  4,  and  ^=4  or  14. 

That  is,  a;4-l=14,  or  4.     y+-=4,  or  14. 

X  y 

Whence,  x=(7dt4jS)  or  (2±V3).    7j=(7^z4j3)  or  (2^:173). 
We  conclude  this  subject  by  the  following  equations : 
Given  6a;5+ll«*— 88a:3— 88a;2+lla:-|*6=0,  to  find  n^  j^qq^^^ 
This  equation  necessarily  has  one  root  equal  to  — 1  ;  therefore 

we  divide  by  (x-\-\),  and  we  find 


ALGEBRA.  165 

Now  5x^^6x—94+--\-^=0. 

X     x^ 

Or.  6(.=  +^)+6(.+l)=94. 

If  we  put  x-\--=7/,  then  x^-{-—=y^^2. 
X  x^ 

And  5(y2— 2)+6y=94. 

^^5       5 

2  1  6y     104 

y+i=±V-    y=4,  or— V 
a;+l=4.     Whence  x=2zizjS. 

X 

1         26 
Or,     x-\--=: — ,  whence  x=  ^,  or  — 5 ;  and  the  five  roots 

X  5 

are  —1,-5,  i,  (2-j-^3),  and  (2— ^S.) 

Given     -S  ('^6i"ll^"^Ql^''M^^'  1-     to  find  at  least  one  of 
the*  values  of  x  and  y. 

By  division,  the  equations  may  be  reduced  to 

^='+^=2/+-  (1) 

And  2,3+J_9(,+i)  (2) 

Now  put  a;+-=P,  and  y4-_=^. 
X  y 

Then    a;3+J_=P3_3p^  and  2/'+— =^='— 3$. 

Substituting  these  results,  and  (1)  and  (2)  become 

P^—3F=Q  (3) 

Q3_sQ=9P  (4) 

Assume  F=nQ ;  then  (3)  and  (4)  become 


164  ROBINSON'S  SEQUEL. 

And  g3_3^^9^^ 

Dividing  by  Q^ 

And                          ^2_3^9„  ,                   (g^ 

From  (5),                n^Q^=i^n+\  (7) 

From  (6),               w^  ^2^(3^+1)3^3  (8) 
Dividing  (8)  by  (7),  gives     3n3  =  l. 

Whence,       9?^=3(9)3  ;  and   substituting  this   in  (6),  and  we 

Lave      ^2^3(9)3-1-3.       Or,      ^=\/3(  9)^+3. 

Having  the  value  of  n  and  Q,  we  have  F=^nQ.      The  values 
of  ^  and  P  will  give  us  x  and  y. 

^7^5.  a:=^(V3+3)^'+(V3— 1)^ 


SECTION    VII. 
mDETERMIIf ATE  ANALYSIS. 


Preliminary  to  this  subject  it  is  proper  to  call  to  mind  a  few 
facts  in  the  theory  of  numbers  ;  for  the  Indeterminate  and  Pio- 
phantine  analysis  is  but  an  application  of  that  theory. 

(1.)     The  sum  of  any  number  of  even  numbers  is  even. 

(2.)     The  sum  of  any  even  number  of  odd  numbers  is  even* 

(3.)     The  sum  of  an  even  and  an  odd  number  is  odd. 

(4.)  The  product  of  any  number  of  factors,  one  of  which  is 
even,  will  be  an  even  number  ;  but  the  product  of  any  odd  num- 
ber is  odd ;  hence, 

(5.)  Every  power  of  an  even  number  is  even,  and  every 
power  of  an  odd  number  is  an  odd  number ;  hence, 

(6.)  The  sum  and  difference  of  any  power  and  its  root  is  an 
even  number.  For  the  power  and  its  root  will  be  both  even,  or 
both  odd,  and  the  sum  or  difference  of  either,  in  either  case,  is 
an  even  number. 


ALGEBRA.  187 


PROPOSITIOJS-S. 

1 .  If  an  odd  number  divide  an  even  number,  it  will  divide  the 
half  of  it. 

Every  number  is  either  odd  or  even — even  numbers  are  in  the 
form  2w,  and  odd  numbers  are  in  the  form  9.n'-\-\.  Now  by- 
hypothesis,  let 

-=0   and  q  a  whole  number. 

Then  27z=^(2//+l) 

It  is  obvious  that  one  factor  in  the  second  member  is  odd, 
therefore  the  other  factor  q,  must  be  even,  otherwise  the  product 
2w  could  not  be  even ;  hence,  q  may  be  expressed  by  2q\  and 

2n=2g'(2n4-l).      Then  —~  ==q\   and  the  odd   number 

^  ^      ^  ^  2n+l       ^ 

(2n'-\-l)  divides  half  the  even  number  2n,  which  was  to  be  dem- 
onstrated. 

2.  ff  a  numher  p  divide  each  of  two  numbers  a  and  b,  it  mil 
divide  their  sum  and  difference. 

a  b       , 

_z=q,      _=g . 

F  P 

That  is  q  and  q\  the  quotients,  are  whole  numbers  by  hypoth- 
esis.    Now  by  addition  we  have  ^  '     =q-\'q',  and  by  subtrac- 

P 
..         a — b  , 

tion,    — —z=q — q. 

P 
But  the  sum  of  two  whole  numbers  is  a  whole  number;  there- 
fore,   (  ^i-  j  is  a  whole  number :  and  it  is  obvious  that^  ) 
is  also  a  whole  number.     Q.  E.  D. 

3.  If  two  members  are  prime  to  each  other,  their  sum  is  prime  to 
each  of  them. 

Let  a  and  b  be  the  two  numbers  prime  to  each  other,  (a+^) 
their  sum,  is  prime  to  each  of  them. 


168  KOBINSON'S  SEQUEL. 

For  by  the  last  proposition  if  («+&)  and  a  have  a  common 
divisor,  their  difference  b  must  have  the  same  divisor ;  but  b  is 
not  divisible  by  a  by  the  supposition ;  therefore,  if  two  numbers, 
&c.     Q.  E.  D. 

Corollary.  In  the  same  manner  it  may  be  demonstrated  that 
if  a  and  b  be  prime  to  each  other,  their  difference  (a — b)  will  be 
prime  to  each  of  them.  ,  '-^^L 

4.  If  two  numbers  arej^rime  to  each  other y  their  sum  and  difference 
will  have  the  common  measure  2,  hut  no  other,  or  their  sum  and  dif- 
ference mil  be  prime  to  each  other. 

Let  a  and  b  be  prime  to  each  other,  their  sum  is  (a-|-6),  and 
difference,  (a — b)  ;  and  if  these  numbers  have  a  common  divisor, 
their  sum  2a  and  difference  25  will  have  the  same ;  but  the  only 
common  divisor  to  2a  and  26  is  2. 

If  one  of  these  numbers,  a  or  b,  be  even,  and  the  other  odd, 
then  (a-j-5)  and  (a — b)  are  both  odd  and  prime  to  each  other. 
For  if  («+5)  and  (a — b)  are  not  prime,  let  them  have  the  com- 
mon measure  n.  Then  by  proposition  2,  n  will  be  the  common 
measure  of  their  sum  and  difference ;  that  is,  the  common  meas- 
ure of  2a  and  25 ;  but  the  only  common  measure  of  these  num- 
bers is  2 ;  therefore,  (a-|-5)  and  (a — b)  are  prime  to  each  other, 
or  have  the  common  measure  2.     Q.  E.  D. 

5.  If  two  numbers  a  and  b  be  prime  to  each  other,  b  being 
the  greater,  then  b  may  always  be  represented  by  the  formula 
|)=;aq-|-r,  in  which  r  is  less  than  &,  and  prime  to  it. 

The  formula  arises  from  the  actual  division  of  b  by  a,  the  re- 
sult is  q,  the  integer  quotient,  and  the  remainder,  r ;  that  is, 

a  a 

Multiplying  this  equation  by  a,  gives  the  formula  ;  r  is  neces- 
sarily less  than  a,  if  we  suppose  q  to  be  the  greatest  quotient. 

We  are  i\ow  to  sbow  that  r  and  a  are  prime  to  each  other.  If 
they  are  not  prime  to  each  other,  they  have  a  common  measure. 


ALGEBRA.  169 

Let  us  suppose  a  common  measure  and  reduce  the  fraction  -  by 

a 

it,  givmg  -  ;  then  the  equation  becomes 
a 

a  a  a' 

But  by  hypothesis  the  fraction  -  is  irrec?wa6^^.     Yet  admittinor 

a  ° 

a  common  measure  to  -,  we  have  the  reduced  fraction  ^~r^_ 


which  is  absurd  ;  therefore,  a  commom  measure  to  -  is  inadmis- 

a 

sible,  and  r  and  a  are  prime  to  each  other. 

6,  If  ayiy  niimher  he  prime  to  each  of  two  others,  it  will  be  prime 
to  their  product. 

Let  c  be  prime  to  both  a  and  b  ;  then  we  are  to  show  that  c  is 
also  prime  to  ab.       * 

By  the  hypothesis  _  is  an  irreducible  fraction  ;  multiply  this 
fraction  by  h,  and  we  shall  have  —  .m 

Now  if  this  last  fraction  is  reducible,  some  common  measure 
must  exist  between  b  and  c ;  but  by  hypothesis  there  is  none ; 
therefore,  ab  and  c  are  prime  to  each  other.     Q.  E.  D. 

Corollary    1.     If  a^=b,  then  -1-= — ;  and  if  c  is  prime  to  a, 

c       c 

it  is  prime  to  a^ ,  a^ ,  and  any  power  of  a. 

Corollary  2.  If  c  is  prime  to  any  number  of  factors  as  a, 
b,  d,  e,  &c.  it  will  be  prime  to  their  product. 

7.  If  two  numbers,  a  and  b,  be  prime  to  each  other,  then  mb 
divided  by  a,  and  m'b  divided  by  a,  will  have  different  remainders 
for  all  values  of  m  less  than  a. 

Let  us  admit  that  the  two  operations  in  division  will  produce 


% 


170'  ROBINSON'S  SEQUEL. 


the  same  remainder, — then  by  performing  the  division  we  shall 
have 

mh        .  r 

—  =9'+- 

*  a  a 

And  —  =  q-\-- 

a  a 


By  subtraction,  -(  m — vi  j=Q — g' 

•Or,  J=  g-g' 


am  —  m 

But  by  hypothesis,    .  is  irreducible,  at  the  same  time  (m — mf) 
a 

is  less  than  a,  which  is  absurd  ;  therefore,  the  two  divisions  can- 
not have  the  same  remainder. 


8.  If  two  numbers,  a  and  b,  be  prime  to  each  other,  the  equation 
ax — by=  ±1,  is  always  jwssible  in  integers.  That  is,  positive  in- 
tegral values  of  x  and  j  mag  be  found  which  will  satisfy  it. 

By  the  preceding  proposition  — =q-\-- ,  and  as  r  may  be  of 

a  a 

any  value  less  than  a,  according  to  the  magnitude  of  m,  r=a — 1 
is  possible.     Whence, 

nih=aq-\-a — 1 
Or,  mb-\-\  =  {q-^\)a 

Now  let  y=w,  and  a?=(5'-|-l)  ;  then      -         ^ 
by-\-\=ax 

Or,  ax — hyz=\ 

But  w  is  a  whole  number,  therefore  its  equal  y  is  a  whole 
number,  and  ($'+1)  is  a  whole  number,  and  consequently  its 
equal  a?  is  a  whole  number  ;  that  is,  ax — by=  1  is  possible,  x  and 
y  being  whole  numbers. 


ALGEBRA.  171 

We  now  come  more  directly  to   the  indeterminate  analysis. 

For  a  perfect  and  definite  solution  of  a  problem,  there  must  bo 
as  many  independent  equations  as  unknown  quantities  to  be  de- 
termined ;  and  when  this  is  not  the  case,  the  problem  is  said  to 
be  indeterminate. 

For  instance,  a;-f-y=20.  Here  x  may  have  any  value  what- 
ever, and  ^he  equation  will  give  the  corresponding  value  toy, 
and  the  number  of  solutions  may  be  infinite;  but  if  we  restrict 
the  values  of  x  and  y  to  whole  numbers,  then  only  19  different 
solutions  can  be  found;  for  x  may  be  equal  to  1,  or  2,  or  3,  &c., 
to  19,  and  y  will  equal  the  remaining  part  of  20. 

In  some  cases,  the  number  of  solutions  is  unlimited  or  infinite. 
ax — hy=c,  represents  a  general  equation  of  the  kind,  and  a  solu- 
tion gives  x=  '  "^  in  which  y  may  be  any  whole  number  what- 
a 

ever  that  will  make  (c-(-5y)=a,  or  any  multiple  of  a ;  but  num- 
berless such  values  of  y  may  be  found,  and  consequently  number- 
less values  of  x. 

N".  B.  Such  equations  are  generally  restricted  to  the  least  values 
of  X  and  y. 

Equations  in  the  form  ax-\-by=c,  may  be  very  limited  in  the 
number  of  their  solutions, — may  have  only  one  solution,  or  a 
solution  may  be  impossible,  when  x  and  y  are  restricted  to  whole 
numbers. 

A  solution  gives  x=-^      ^ . 
a 

Now  if  c  is  very  large,  and  b  and  a  small,  y  may  take  a  great 

number  of  integral  values,  before  the  numerator  becomes  so  small 

as  not  to  be  divisible  by  a. 

If  we  make  y=l,  and  then  find  that  i \  is  a  proper  frac- 
tion, a  solution  is  impossible  in  integers. 

The  equation  ax-\-hy=^c  is  always  possible  in  integers,  when  e 
is  greater  than  {ab — a — b),  and  a  and  b  prime  to  each  other. 

The  equation  ax-\-by=c,  is  possible  sometimes,  or  rather  with 
tome  numbers,  when  c  is  less  than  (a5 — a — b).  For  example, 
7a:-{-13y=71,  is  impossible  in  integers,  because   (7-13 — 20)  is 


172  ROBINSON'S  SEQUEL. 

not  greater  than  71.  But  7x-|-13y=27,  is  possible,  giving  rr=2, 
and  y=l.  '  That  is,  7x-\-13i/ iput  equal  to  any  whole  number 
greater  than  71,  will  admit  of  a  solution  in  integers;  and  put 
equal  some  numbers  less  than  71,  will  admit  of  a  solution. 

In  all  these  equations  a  and  b  are  supposed  to  be  prime  to  each 
other.  If  they  have  a  common  divisor,  that  same  divisor  must 
divide  c,  .or  a  solution  is  impossible  in  integer  numbers. 

For  if  «ir4-Jy=-,  it  is  obvious  that  ax  is  a  whole  number,  also 
n 

by  is  a  whole  number,  and  the  sum  of  two  whole  numbers  can 
never  be  equal  to  an  irreducible  fraction,  as  the  preceding  in- 
dicates. 

For  a  particular  example,  6a;-|-9y=32,  is  impossible  in  whole 
numbers.  Dividing  by  3,  and  2.c-l~%=  V-  ^^  ^  ^^  ^^  ^^  ^ 
whole  number,  2a;  must  be  a  whole  number,  and  3y  must  also  be 
a  whole  number  ;  but  it  is  impossible  for  any  two  whole  numbers 
to  be  equal  to  \^ . 

In  cases  where  solutions  are  possible,  our  rules  of  operation 
rest  entirely  on  the  following  facts  : 

1st.  A  whole  number  added  to  a  whole  numtber ^  the  sum  is  a  whole 
number, 

2d.  A  whole  number  taken  from  a  whole  number,  the  remainder  is 
a  whole  nurriher. 

3d.  Multiply  a  whole  number  by  a  whole  number,  and  the  product 
is  a  whole  number. 

EXAMPLES. 

( 1 .)  Given  3a:-j-5y=35,  to  find  the  values  of  x  and  y  in  whole 
numbers. 

35— 5y 

3 

Because  x  must  be  a  whole  number,  the  fractional  form ^ 

3 

must  be  a  whole  number,  or  (11— y)+ — ~-  must  be  a  whole 
number.  But  (11 — y)  is  obviously  a  whole  number  ;  therefore, 
the  other  part,  ( -ZIJ^.  )  must  be  a  whole  number  also. 


ALGEBRA.  173 

Again,  as  y  is  a  whole  number,  -Z  is  in  fact  a  wliole  number, 

o 

which  added  to  {  -Zl-^  ),  and  the  sum  is  -^-,  a  whole  number. 
V     3     /  3 

In  this  last  expression  the  coefficient  of  y  is  reduced  to  1  under- 
stood, and  the  operation  had  that  end  in  view. 

Put  this  last  expression  equal  n  ;  that  is  any  wliole  number,  or 
rather  some  whole  number. 

^+l=n, 
3 

Or,  y=3^^— 2. 

In  this  last  equation  we  can  take  n=^0,  1,  2,  3,  <fec.,  as  far  as 
such  substitutions  will  correspond  to  the  values  of  x. 

If  we  take  w=0,  then  y= — 2,  an  inadmissible  result ;  for  we 
demanded  positive  values.  Therefore,  we  take  7^=l,  then  y=l, 
and  a;=10.  If  7i=2,  then  y=4,  and  a?=3.  If  ?^^3,  y=7,  and 
a;=0.  Hence  the  last  is  not  a  full  solution,  and  the  equation  only 
admits  of  two  solutions  ;  namely,  .r=10  or  3,  and  y=l  or  4. 

(2.)  Given  2>5x — 24y=68,  to  find  the  least  values  of  x  and  y 
in  whole  numbers. 

We  require  the  least  values,  because  an  unlimited  number  of 
solutions  may  be  found. 

From  the  equation,  x=^^'Ay=.l+^l±l^ . 
33        •       '        35 

Hence,        '"     -=  some  whole  number:  but — •^=  some  whole 
35  35 

number ;  therefore,,  by  taking  the  difference  of  these  two  whole 

numbers  we  have  — ^    — =  some  whole  number. 
35 

Three  times  a  whole  number  is  a  whole  number ;  therefore, 

33y— 99     33y— 29     ^  ,    ,  ,     \ 

— = — — 2=  some  whole  number.* 

35  35 

— t. =iwh.     Also,  — ^-=iwh. 

35  35 

*Subsequently  we  shall  put  wh  to  represent  the  phrase,  some  whole  number. 


174  ROBINSON'S   SEQUEL. 

Whence,      ^^y^^lyi=??.^^y+^=wh. 

35  35  35 

\     35     /  ^       "^    35 

Having  thus  deprived  y  of  its  numerical  coefficient,  we  may 

put  ^-"t^-?=w.     Whence,   y=35w— 32. 

Taking  w=l.     y=3,  and  x= — 3" =4,  the  least  possible 

35 

values  of  x  and  y  in  integers,  as  was  required. 

The  next  values  are  ic=28,  and  y=38. 

(3.)  A  man  proposed  to  lay  out  $500  for  cows  and  sheep  ;  the 
cows  at  the  rate  of  '^  17  per  head,  and  the  sheep  at  ^2.  How  iimny 
of  each  could  he  purchase  ? 

Let  a;=  the  number  of  cows,  and  y=  the  number  of  sheep. 

Then  17a:-|-2y=500,  is  the  only  equation  that  can  be  obtained, 
and  X  and  y  must  be  whole  numbers  by  the  nature  of  the  prob- 
lem— he  could  not  purchase  a  part  of  a  cow,  or  a  part  of  a  sheep. 

To  find  the  least  number  of  cows,  transpose  17a;. 

Then  y=250— 8a;— ^. 

Now  as  X  and  y  must  be  whole  numbers,  _  must  be  a  whole 
^  2 

number,  and  the  least  number  corresponding  to  x   must  be  2 ; 

corresponding  to  this,  y=233. 

Under  all  suppositions,  the  number  of  cows  must  be  divisible 
by  2. 

Now  if  the  object  was  to  purchase  as  many  cows  and  as  few 
sheep  as  possible,  we  would  transpose  the  other  term  thus  : 

:.=522=:?^=:29+ZlI?^ 
17  ^    17 

Whence,  !=I?^=im,'A.     Or,  ^Zl^l^wh  ;  to  this  add 

17  17 

III,  and  ^^==3+^±^ 
17  17  ^   17 


ALGEBRA.  176 

Therefore  JA^=zn,  or  y=\ln — 6. 

,     17 

Put  w=l,  then  y=12,  the  smallest  number  of  sheep.  The 
corresponding  value  of  a:=28. 

The  number  of  cows  may  be  any  one  of  the  even  numbers 
from  2  to  28. 

(4.)  A  man  wished  to  spend  100  dollars  in  cows,  sheep,  and 
geese ;  cows  a^  10  dollars  a  piece,  sheep  at  2  dollars,  and  geese  at 
25  cents,  and  the  aggregate  number  of  animals  to  he  100.  How  many 
must  he  purchase  of  each  ? 

Let  x=.  the  number  of  cows,  y  the  sheep,  and  z  the  geese. 

Then  10a;+2y+-=100.  (1) 

And  a:+y4-2=100.  (2) 

Clear  equation  (1)  of  fractions,  and 

40a;4-8y-(-s=400. 
x-\.  ^+0=100. 

39a;+72/      =300. 

ir= £l=7-[- —       ^=  a  whole  number. 

39  '      39 

Or,  6( — IZIJl  )= H — ^=  a  whole  number;  add  — ^  and 

\     39     /  39  39 

4H-135    ^^  40y+1350  H:24^  ^^^^^^  ^^^^^^^ 

39  39  '^     ~    39 

Therefore,        y+24_^      ^^^  y_39p_24_i5, 

09  r 

This  value  of  y,  gives  ar=5.     Hence,  2=80. 

If  we  take  ^=2,  we  shall  have  y=z5A\  then  a:  will  come 
a  minus  quantity,  an  inadmissible  circumstance  in  any  problem 
like  this.  Therefore,  5  cows,  15  sheep,  and  80  geese,  is  the  only 
solution. 

(5.)  A  person  spent  28  shillings  in  ducks  and  geese  ;  for  the 
geese  he  paid  4s.  Ad.  a  piece,  and  for  the  ducks,  25.  Qd.  a  piece. 
What  number  had  he  of  each  ? 

Let  a;=  the  number  of  geese,  and  y  the  number  of  ducks. 


% 


176  ROBINSON'S  SEQUEL. 

28-12. 


Then     52ar+302/=28-12.     Or,    26x-\-l52/=:168. 

,  3— lire 


15 


\  Whence,  =wh.     But =wh. 

16  15 

By  addition,     C-^+^)4=wk.    l^Hd?=.+f+15 
^  \    15    /  15  ^15 

_JL — z=n.     x=15n — 12.     a;=3,  when  w=l. 
15 

Then   ?/=8— 2=6.     When  n=2,  x=lQ,  and  y=—20.     But 

this  is  inadmissible;  therefore,  x=3  and  y=6,  is  the  only  possible 

solution. 

(6.)  Divide  the  number  100  into  two  such  parts  that  one  of  them 
may  be  divisible  by  7,  the  other  by  11, 

Let  7x=  one  part,  and  lly  the  other  part. 
Then  7a:+lly=100. 

_100— 7a;__        1— 7a; 

^       IT""     """Tr" 

11    ^11        11  U  '    11 

^+?=?i.     x=lln—3. 
11 

Now  put  ?z=l,  then  a;=8  and  y=4.  56  and  44  are,  therefore, 
the  required  parts. 

(7.)  Mnd  the  least  number  which  being  divided  by  6  shall  leave 
the  remainder  3,  and  being  divided  by  13  shall  leave  the  remainder  2. 

Let  iV=  the  number  sought,  and  x  and  y  the  quotients  arising 

from  the  divisions. 

•^V— 3  .  JV—2 

Then  =x,  and  =y 

6  13       ^ 

Whence  ^^=6x+3,  and  iVi=  13y+2. 

Consequently,      6z-\-l  =  lSy. 

6  6 


ALGEBRA.  HT 

Then  ^~-=n,  or  y=z6n-\-l 

For  the  least  values  of  y  we  must  take  w=0,  then  y=  1  and 
x=2.     ButiV=6;r+3=15.  V^ 

We  may  determine  iV"  more  directly  without  the  aid  of  x  and         ft 
y,  thus : 

.  =  some  whole  number  ;  also,  =wh. 

6  13 

As  iV  does  not  contain  a  coefficient  to  be  worked  off,  we  may 
7\r    3 

put    =^,  and  iV^=6p-["2  i^  which  any  integral  value  put  for 

6 

J)  will  satisfy  the  first  whole  number,  but  the  other  must  be  sat- 

isfied  also  ;  then  put  the  value  of  N  in ;  that  is, 

^  13 

^^      I-=  some  whole  number. 
13 

13  13  13  13  13 

Whence,  p=1i3q-\-2.  Assume  2'=0,  then ^=2  and 
JV=6i?+3=15,  as  before. 

(8.)  What  number  is  that  which  being  divided  by  11,  leaves  a 
remainder  of  3,  divided  by  1 9,  leaves  a  remainder  of  5,  and  divided 
by  29,  shall  leave  a  remainder  o/"  10  ? 

Let  ^y  be  the  required  number,  and  x,  y,  and  z  the  several 
quotients,  and  of  course  they  must  be  whole  numbers. 

Then  llar+3=iV;  and  19y4-5=iV,  and  292+10=iV. 

Hence,     ir=??!±I,  and  x='^?li^.     19^=29^+5 

29^+5^    .  lOH-5 

19  ~     19     * 

200+10        ,         0+10 

! — =zwh,  or  -J — =n.     z=19n — 10. 

19  19 

Any  integer  written  in  place  of  n  will  give  integer  values  to  z 
and  y,  but  x,  or JLL  must  be  a  whole  number  also. 

Hence  such  a  value  of  n  must  be  found  as  will  make 
29(19w— 10)+7        , 


m  ROBINSON'S  SEQUEL. 

Or,      561^290+7^651.^-283^^^^         n-8^^^ 
11  11  ^  11 

^ o 

Whence,      . =p.     w=  11^+8. 

For  the  least  value  of  n  put^=0  ;  then  w=8. 
But  2=19n—10=19'8— 10=162— 10=142. 
And  iV^=29- 142+10=4128,  the  number  required. 

(9.)     Required  the  least  nuv}her  that  can  he  divided  by  each  of  the 
nine  digits,  without  remainders. 
Let  xz=i  the  number. 

Then   -,  -,   -,  -,  -,  -,  -,  -,  must  all  be  whole  numbers. 
2     3     4     6     6     7     8     9 

Now  if  we  make  -  a  whole  number,  _  and  _ ,  the  double  and 
8  4  2 

C|[uadruple,  will  be  whole  numbers  of  course.   Also,  if  -  is  a  whole 

number,  —  will  he  a  whole  number. 
3 

Therefore,  we  have  only  to  find  such  a  value  of  x  as  will  make 

_,  _,  _,  _,  -whole  numbers.     -,   may  also  be  cast   out,  on 
9     8     7     6     5  6  -^ 

consideration  that  6=2-3,   and  2*3  are  factors,  one  of  9,   the 

other  of  8,  in  the  preceding  expressions. 

Hence  we  have  only  to  find  such  a  value  of  x  as  will  make  each 

of  the  fractions  -,  _,  -,   and  -whole  numbers;  and  as  these 
9     8     7  6 

denominators  contain  no  common  factors,  their  product  is  the 
least  number  that  will  answer  the  condition. 
Whence,      ir=72  •  36=2520. 

(10.)  A  market  woman  has  some  eggs,  which  when  counted  out 
by  twos,  threes,  fours,  and  fives,  still  left  one ;  hut  when  counted  hy 
sevens,  there  was  none  left.  Wliat  was  the  least  possible  number  of 
eggs  she  could  have  had  ?  Ans.  301. 

X——\       X       1       X       1       X       1  1   X 

This  problem  requires ,  ,  ,  ,  and  _  to  be 

whole  numbers. 


ALGEBRA.  1^9 

j  is  a  whole  number,  / j ,  its  double,  must  be  a 

whole  number  of  course ;  hence,  we  have  only  to  make 

,  .JUL- ,  ,  and  --  whole  numbers. 

3  4  5  7 

Put  the  first  expression  equal  to  the  whole    number  n  ;  then 

a:=3w-[-l.     This  will  satisfy  the  first  expression  ;  but  the  others 

must  be  satisfied  also;  therefore,  substitute   (3^-|-l)   for  x  in 

them. 

Then  — ,    — ,   and  — X-,  must  be  whole  numbers. 
4       5  7 

VS/i        7     6w         .  n        ^         n        ■, 
5  5  ^5  5 

That  is,  n=bp  ;  and  substituting  this  value  of  n  in  the  other 
two  expressions,  we  have 

— ^ ,  and     ^~^   ,  to  be  made  whole  numbers. 
4  7 

]^P+lz=2pJf-Pj^  =:wh.     Whence,  put  ^±1=^.     p=lq--\. 

15/?       . 
Finally,  this  value  of  p  put  in  — ^ ,  gives 

4 

im ,  which  must  be  made  a  whole  number 

4 

before  we  can  be  sure  of  a  result  which  will  satisfy  all  the  con- 
ditions, 

l^^~^=(26^-3)+?=?=...A. 

Whence,  'Zl~-?=/.     y=:4«:+3. 

For  the  least  value  of  q,  put  /=0 ;  then  $-=3. 

Butjt?=7^— 1=20.     w=5p.     Then?*=100.     .i-=3;i-f  1=301. 

(11.)  Required  the  year  of  the  Christian  Era  in  which  the  solar 
cycle  was,  or  will  be  15,  the  lunar  cycle  12,  and  the  Roman  Indiction 
12. 

N.  B.  The  operator  must  of  course  know  the  exact  import  of 
these  terms,  and  the  facts  in  the  case,  before  he  can  be  required 
to  solve  the  problem. 


180  ROBINSON'S   SEQUEL. 

The  solar  cycle  is  a  period  of  28  years,  at  the  expiration  of 
which,  the  days  of  the  week  return  to  the  same  days  of  the 
month,  (provided  a  common  centurial  year  has  not  intervened.) 

The  first  year  of  the  Christian  Era  was  the  tenth  of  this  cycle ; 
therefore,  we  must  add  9  to  the  year  and  divide  the  sum  by  28, 
and  the  remainder  will  be  the  number  of  the  cycle. 

The  lunar  cycle,  or  Golden  number,  as  it  is  sometimes  called, 
is  a  period  of  19  years,  after  which  the  eclipses  return  in  the  same 
order  as  in  the  previous  19  years.  The  first  year  of  the. Christ- 
ian Era  was  the  second  of  this  period ;  therefore,  we  must  add  1 
to  the  year  and  divide  by  19,  and  the  remainder  is  the  year 
of  the  lunar  cycle. 

The  Roman  Indiction  is  not  astronomic.  It  is  a  period  o€  15 
years,  the  first  of  our  Era  being  the  4th  of  the  Indiction  ;  there- 
fore, add  3  to  the  year  and  divide  by  15,  and  the  remainder  is 
the  Indiction. 

The  reader  is  now  prepared  to  solve  this  or  any  other  similar 
problem. 

Let  X  represent  the  required  year  ;  then  the  problem  requires 
that 

a;4-9— 15     a;-fl— 12     a:-f-3— 12 

28        '  19        '  15 

/^ g  ^ ]  \  /J 9 

should  be  whole  numbers  ;  that  is, , , ,  must  be 

28  19  15 

whole  numbers. 

The  first  expression  will  be  satisfied  by  putting  it  equal  to  any 
whole  number  n\  then  a;=28w-f-6. 

But  the  other  two  expressions  must  be  satisfied  also. 

Therefore  ^^MltU  and  ^Mlt:^ 

19  15 

Or,  '^''^n—b  ^^^  28?i— g  ^^^^  ^^  ^j^^j^  numbers. 

19  15 

Or.  ^^~^  and  ^^^~^  must  be  whole  numbers. 

19  15 

157^_13n-3_27^+3_^^,;^       7(2n+3)_^^^ 

15  15  15  15 

15n_14n+21_^^,;^       n-21 

16  16  *         16 


ALGEBRA.  181 

Whence,      n=15p-\-21.      Every  expression  is  now  satisfied, 
I — 
19" 


except    ^        ;  to  satisfy  this,  write  in  the  value  of  n  ;  then 


9(15;.+21)-5_,^ 
19 

19  ^^  ^     19 

Whence,      ^+^i=t.A.,  and  ^5H^?2=^+6+^-H6 
19  19  ^  ^    19 

19        ^      ^         ^ 
We  canhot  take  q  less  than   1 .     The  least  value  of  p  will  then 
be  3.     But  ?2=15p+21=66. 

a;=28?i-|-6=28-66+6  =  1854. 
If  no  interruption  was    to   be   made   by  the   centurial  years, 
the  coincidence  of  these  cycles  would  not  occur  again  until  the 
year  9834,  which  we  find  by  making  q=2. 


SECTION   VIII. 

TO  DETERMINE  THE  NUMBER  'OF  SOLUTIONS  AN  EQUATION 
IN  THE  FORM  AX-{-BY=C  WILL  ADMIT  OF. 

An  equation  in  the  form  ax — hy=c,  will  admit  of  an  unlimited 
number  of  solutions,  because  x  and  y  can  increase  together;  but 
not  so  with  equations  in  the  form  ax-[-by=c  ;  for  in  them  an  in- 
crease of  X  will  cause  a  decrease  of  y,  and  an  increase  of  y,  a 
decrease  of  x;  but  neither  x  nor  y  are  permitted  to  fall  below  1. 
If  c  is  very  large  in  relation  to  a  and  b,  the  equation  may  have 
a  great  number  of  solutions,  and  we  are  now  about  to  show  a 
summary  method  of  determining  the  solutions  in  any  given  case. 

The  equation  ax' — hy'=\  is  always  possible  in  wjiole  num- 
bers, (Prop.  8,  sec.  vii;)  therefore,  c  times  the  equation  is  also 
possible ;  that  is, 

acx' — cby'z=c,  is  possible. 

But  ax-\-by=^c,  is  a  general  equation. 

Put  these  two  values  of  c  equal  to  each  other,  then 


182  ROBINSON'S  SEQUEL. 

(xx-{-byz=zacz — cby'  ( 1 ) 

From  (1),  x=cx—(^yjh\b.  (2) 

For  the  sake  of  perspicuit}-,  put  ^  "1"^=^. 

a 

Then  (2),  becomes         x^cx' — hn  (3) 

Multiply  (3)  by  a,  and  substitute  the  value  of  ax  in  (1), 
Then  acx — abm-\-by=acx' — cby' 

Whence,  yz=iam — cy'  (4) 

From  (3),  we  find  7^=^:-— _  (6) 

h       b 

From  (4),  we  find  m=^]^+l  (6) 

a      a 

Equations  (5)  and  (6)  show  that  m  must  be  greater  than  ^ 

a 

and  at  the  same  time  less  than  — 

b 

Therefore,  the  limits  to  m  are  found. 

Now  let  us  observe  equations  (3)  and  (4)  ;  rr  must  be  a  whole 
number,  and  as  c,  x,  and  b  are  whole  numbers,  m  must  be  taken 
in  whole  numbers,  and  it  may  be   any  whole  number  between 

a  b 

The  number  of  solutions  will,  therefore,  correspond  with  the 

difference  between  the  integral  parts  of  the  fractions  —  and  ^ 

h  a 

except  when  ^  is  a  whole  number,  in  that  case  x  becomes  b,  and  - 
/  b  b 

must  be  considered  a  fraction,  and  rejected.  If,  however,  we  in- 
tend to  include  0  among^  the  integral  values,  this  precaution  need 
not  be  observed. 

EXAMPLES. 

(1.)  Required  the  number  of  integral  solutions  of  the  equation, 
7a;+9y=100. 
'    First  find  the  least  value  of  x'  and  y'  in  the  equation 
7a;'— 9/=  1. 


ALGEBRA.  18S 

The  result  will  be  x'—4.     y'=3. 

Then  -'=l?2:!=44l    'l='^^,=42l 

b  9  9      a  7  7 

Disregarding  the  fractions,  the  difference  of  the  integral  parts 
is  2,  showing  two  integral  solutions  to  the  equation. 

If  we  had  taken  the  difference  between  ^^^  and  ^^,  thus ; 
B|ofli__2  7.oo_i|_i^  ^e  might  have  come  to  the  conclusion  that 
the  equation  would  admit  of  only  one  integral  solution. 

The  integral  difference  in  this  case  is  not  the  difference  of  the 
integrals. 

When  the  fractional  part  of  —  is  not  less  than  the  fractional 

b 

part  of  —,  but  equal  or  greater  than  it,  we  can  find  the  number 
a 

of  solutions  by  taking  the  difference,  thus  — — ^=^i^f yj. 

b       a  ab 

But  ax' — by'=\  ;  therefore,  — — Jl.= — 

b       a      ab 

Whence  we  conclude  that  —  will  in  this  case  show  the  num- 
ab 

ber  of  solutions. — In  all  cases  it  will  be  the  number,  or  one  less. 

(2.)  What  number  of  integral  solutions  will  the  equation 
9;5+13y=2000  admit  of?  Am.  17. 

9-13=117)2000(17. 

That  is,  we  are  sure  of  17  different  solutions,  and  there  may 
be  18. 

The  equation  bx-\-9y^=40  admits  of  no  solution  in  whole  num- 
bers, c,  40,  is  not  divisible  by  5* 9=45,  that  is,  no  unit  in  the 
quotient.     Yet  the  equation 

5a;-|-9y=:37,  admits  of  one  solution. 

The  auxiliary  equation  5x — 9y'=\,  gives  a;'=2,  and  y'=l. 


Therefore,  ^^J^=^.    ^=?Z=7i 


■2. 

b       9        ^       a       5        '* 


Here  the  difference  of  the  integrals  is   1,  indicating  one  solu- 
tion.    In  fact  x=2,  y=S. 

How  many  solutions  will  the  equation  2x-\-5g=40  admit  of? 


184  ROBINSON'S  SEQUEL. 

The  auxiliary  equation,      2a;' — 6/=l,  gives   x'=S,  y'=l. 

—=24.    ^=20.     Or,  4  solutions. 
^  b  a 

But  observe  that  —  in  this  case,  is  a  complete  integral,  24  ; 
b 

according  to  previous  considerations,  we  must  deduct  one,  and 

the  number  of  solutions  are  3,  as  follows  :  x=5.  10.  16.    y=6. 

4.  2,  and  no  other  solution  can  be  found. 

(3.)  What  number  of  solutions  in  whole  numbers  can  be  found 
for  the  equation  3x-\-5y-\-7z=100. 

As  X  and  y  each  cannot  be  less  than  one,  z  cannot  be  greater 
than  ^^^^^~^=13}.  That  is,  z  cannot  be  greater  than  13,  in 
whole  numbers.  Now  suppose  0=1,  and  the  equation  becomes 
3a:-|-5y=93. 

The  number  of  solutions  for  this  equation,  found  as  previously- 
directed,  is  6.     That  is    j  x  =  26.  21.    16.   11.     6.     1. 
(  y=z    3.     6.     9.    12.    15.    18. 

Now  X  and  y  can  make  these  six  changes,  and  z  be  constantly 
equal  to  1,  and  satisfy  the  primitive  equation. 

We  may  observe  here  that  x  diminishes  from  one  solution  to 
another  by  the  coefficient  of  y,  and  y  increases  by  the  coefficient 
of  X,  but  this  is  not  a  general  principle. 

Take  2=2,  and  the  equation  becomes  3a;-|-5y=86. 

This  equation  has  also  6  solutions,  z  being  through  all  the 
changes  of  x  and  y  equal  to  2. 

Now  take  s=3,  then  the  original  equation  is  3x-\-5y=^lQ,  This 
equation  has  five  solutions. 

Now  take  0=4,  then  3a;-|-5y=72.  This  equation  has  four 
solutions. 

Take  0=5,  then  3a;-[-5y=66.    This  equation  has  four  solutions. 

Take  0=6,  then  3ar-}-6y=58.    This  equation  has  four  solutions. 

In  this  manner,  by  taking  0  equal  to  all  the  integers  up  to  12  in 
succession,  we  find  41  solutions  to  the  primitive  equation. 

(4.)     Required  some  of  the  integral  solutions  to  the  equation 
14a;+19y+2l0=252. 


ALGEBRA.  186 

Here  we  observe  that  14  and  21  and  252,  are  all  divisible  by  7, 
therefore  y  must  be  7,  or  one  of  its  multiples  ;  suppose  it  7,  and 
divide  the  whole  by  7. 

Then  '      2^+19-1-32=36. 

Or,  2^+30=17 

Whence,  x  may  equal  1,  and  2=5,  or  ^=7  and  2=1.  Or, 
fl;=4  and  2=3. 

ix—1.     4.     1. 
>      Hence,   \  y=7.     7.     7. 
(2  =  1.     3.     5. 
As  these  examples  are  of  little  practical  utility,  we  give  no 
more  of  thep. 


SECTION    IX. 


DIOPHAI^TINE  ANALYSIS. 


Diophantus  was  a  Greek  mathematician,  who  flourished  in  the 
early  days  of  science  :  and  the  analysis  that  bears  his  name, 
mostly  refers  to  squares  and  cubes. 

The  object  of  this  analysis  is  to  assign  such  values  to  the  un- 
known quantities  in  any  algebraic  expression,  as  to  render  the 
whole  a  square  or  a  cube,  as  may  be  required. 

The  first  principles  of  this  branch  of  science  are  very  simple, 
but  in  their  application,  they  expand  into  the  region  of  impossi- 
bility. 

To  Euler  and  Lagrange,  we  are  indebted  for  most  that  has  ap- 
peared on  this  subject. 

Case  1st.  The  most  simple  expression  to  be  made  a  square, 
is  of  this  form  :  '- 

ax-\-h. 

All  we  have  to  do,  is  to  put  this  expression  equal  to  smne  square, 
say  n^  ;   then 

a;= ,  and  n,  a,  and  &may  be  taken  at  pleasure. 

a 

The  result  will  give  a  value  of  x  which  will  render   (aa;+5),  a 

square  as  required. 


.*.. 


•  * . 


^w 


186  ROBINSON'S   SEQUEL. 

I  EXAMPLES. 

(1.)    Three  times  a  certain  number  increased  by  10,  is  a  square. 
What  is  the  number  ? 

Here   a=3,   ^=10,  and  fl;= — III — 

o 

If  we  put  w=l,  then  x^= — 3,  and  ax-\-b^= — 9-}- 10=1,  a  square. 

If  we  put  w=2,  then  ic= — 2,  and  aar-(-5=4,  a  square;    and  by 

taking  different  values  for  n,  we  can  find  as  many  squares  as  we 

please. 

(2.)     Find  such  values  of  x  as  will  render  the  following  expres- 
sions squares  : 

(9a:+9),  (7;r+2),  (3;r— 5),  (2^^— i). 
All  these  expressions   correspond  to  the  general  expression, 
[ax-^b.) 

Case  2d.     To  advance  another  step,  we  require  such  values 
of  X  as  will  render  any  expression  in  the  form 

{ax^-\-bx)  a  square. 

Because  a:  is  a  factor  in  every  term  of  the  power,  we  will  make 
it  a  factor  in  the  root :  that  is,  put  the  root  equal  nx ;  then 
ax^  -\-bx=^n^  x^ . 
ax-\-b=n'^x. 

x=-A- 
n^ — a 

^    *  EXAMPLES. 

(3.)     Six  tim£s  the  square  of  a  certain  number,  added  to  five  times 
the  same  number,  is  a  square.      What  is  the  number  ? 

Ans.  x= ,   that  is,  the  number  is  expressed  by 

with  the  liberty  to  give  any  value  to  n  that  we  please. 

If  we  make   n=l,  then  a;= — 1,  and  (ax^-{-bx)  will  become 
1,  a  square  as  required. 

If  we  make  w=2,  then  a;=_-|= — f,  and  ax^ -\-bx=6.%^ — ^^=2by 
a  square  as  required,  and  many  other  results  may  be  obtained. 


ALGEBRA.  187 

(4.)  Find  what  values  of  x  will  render  the  following  expressions 
squares : 

(5a?2— 3a;),  {lx^—\bx),  {\2x^—yx),  {yx'^—^x.) 

(5.)  Find  such  a  number  that  if  its  square  he  divided  by  12,  and 
one-third  of  the  number  be  taken  from  the  quotient,  the  remainder  will 
he  a  square  numher. 

Ans.  16  is  one  number,  and  there  are  many  other  numbers 
that  will  correspond  to  the  conditions. 

The  practical  utility  of  this  analysis  may  be  exemplified  in 
forming  examples  in  quadratic  equations. 

Thus  ax^ — hx=zN,  is  a  quadratic,  and  we  would  assign  such 
values  to  N,  as  will  make  the  values  of  x  commensurable  quan- 
tities. 

h         b^  h^ 

Completing  the  square,  gives    x^ — -x-\- =iV'4- . 

a       4a ^  4a^ 

To  make  the  values  of  x  -commensurable,  it  is  necessary  to 

/          5^  \ 
make  the  expression  t  A^-|- )  a  perfect  square,  and  this  we 

can  do  by  putting  it  equal  to  some  definite  square,  (by  case  1st  a 
and  b  being  known  quantities,)  and  deducting  the  values  of  N. 

Case  3d.  Expressions  in  the  form  [x^d=.ax-^h),  can  be  made 
perfect  squares,  by  putting  them  equal  to  the  square  of  [x — n.) 

This  hypothesis  assumes  [x — n)  to  be  the  square  root  of 
{x^diziax-\-b),  and  as  this  expression  may  be  any  number  between 
zero  and  infinity,  we  now  enquire  whether  it  be  possible  that 
[x — n)  can  always  represent  the  root,  whatever  it  may  be.  We 
reply,  it  can. 

If  X  is  large  and  n  small,  [x — n)  will  be  large.  If  x=in,  then 
[x — n)  will  be  zero.  If  n  is  numerically  greater  than  x,  then 
(x — n)  will  be  negative  ;  but  the  square  of  a  negative  quantity  is 
positive  ;  therefore,  n  can  be  so  assumed  that  [x — nY  can 
equal  to  any  positive  quantity  whatever.  -  ::^ 

That  is,  x'^±:ax-{-b=x'^—2nx-\-n^.  "* 

Or,  x:= 

2w±a 

An  expression  in  which  n  may  be  taken  of  any  value  whatever, 

and  we  shall  have  a  corresponding  value  of  x. 

_  ^  '*•*  i^ 


% 

♦ 


^^^*v,  > 


188  ROBINSON'S  SEQUEL. 

Case  4th.     An  expression  in  the  form  (ax^  zhhx-\^c^  ) ,  can  be 
made  a  complete   square,  by  assuming  its   square  root  to  be 

(c — 71X.) 

Because  c^  is  in  the  power,  c  must  be  in  the  root,  and  it  is  obvi- 
ous that  X  must  be  a  factor  in  the  other  part  of  the  root. 
^'  Whence,  ax^ztbx-\-c'' =c'^—2cnx-{-n^x'^. 

•■^^  axd3=—2cn-\-n^x. 

2cn±:b 


EXAMPLE. 

What  value  shall  be  given  to  x  to  make  8x^-[-17x-|-4  a  perfect 
square  ? 

Here  a=8,  5=17,  c=2. 

Whence,  :.=  ^^+^^. 

If  7^=l,  then  x——'2>,  and  '8;c2-|-17;c+4=25,  a  square.  If 
n=3,  then  2;=29,  and  the  value  of  the  expression  is  7225,  the 
square  of  85. 

Case  5th.  An  expression  in  the  form  {^ax"^ -\-bx-\-c) ,  in  which 
neither  the  first  nor  the  last  term  is  a  square,  neither  branch  of  the 
root  can  be  taken,  and  the  expression  cannot  be  made  a  square, 
unless  we  can  separate  it  into  two  rational  factors,  or  unless  we 
can  find  some  square  to  subtract  from  it,  which  will  leave  a  re- 
mainder susceptible  of  being  separated  into  two  rational  factors. 

By  placing  the  expression  [ax^ -\-bx-\-c)  equal  to  zero,  and 
solving  the  quadratic,  we  shall  have  two  factors  of  the  expression, 
but  whether  rational  or  commensurable  factors  or  not,  is  the  subject 
of  enquiry. 

,  To  find  the  factors  which  make  the  product   ax^  -\-bx-\-c,  i^Mt 

this  expression  equal  to  0,  and  work  out  the  values  of  x  thus, 

ax^-^-bx-^c^O.     Or,   ax^-\-bx^= — c.     Complete  the  square,  and 

4:a^x^-{-Aabx-\-b^=b'^—^ac. 

Or,  ^ax-\-b=±J{b^—'Aac.) 

Or  ^=±iV(*'— 4ac)— A. 

2a^^  ^     2a 


/ 


ALGEBRA.  '*  189 

We  now  perceive  that  the  values  of  x  must  be  rational,  previ- 
ded    J{b^ — 4ac)  is  a  complete  square.     If  it  be  so,  let 

J_  /(P_4ac) ^=m,  and  —}_J{b^—4ac)—^=n. 

Then  the  two  values  of  a;  are  x=m  and  x=n,  and  (x — m)(x — n), 
are  the  factors  which  will  give  the  expression  ax^-^x-^-c. 

EXAMPLES. 

(1.)  Find  such  a  value  ofxas.  mil  make  6x^-|-13x-|-6  a  square. 

Here  a=6,  5=13,  c=z6.  h^  =  l69,  4ac=144,  P—4ac=25, 
and  J(b^—4ac)=5.  Now  12a:+13=db5.  a:=— |.  Or,  x=—^. 
Or,  3;i:+2=0,  and  2a;+3=0. 

That  is,  (Sx-\-2)  (2a:-|-3),  will  produce  the  expression 
6a;2+13:r-f-6. 

Now  to  find  such  values  of  x,  as  will  make  the  expression  a 
square,  put 

(3a:+2)  (2x-{-3)=n^3x-\-2y . 

That  is,  take  either  factor  of  the  expression  for  a  factor  in  its 
square  root. 

Then  2x-{'S:=n^3x-{-2.) 

2^2—3 

x= 

2—3n^ 

Take  n=l,  then  x=l,  and  the  expression  becomes  ,  % 

6-|- 134-6=25,  a  square. 

Take  n=2,  then  x= — i,  and  the  expression  becomes  1,  a 
square. 

Take  n=S,  then  x= — f,  and  the  expression  becomes  -^j,  a 
square. 

(2.)  Find  such  a  value  of  x  as  shall  render  the  expression 
(\Sx^-{-15x-{-7),  a  square. 

Here  as  neither  the  first  nor  the  last  term  is  square,  nor  (b^ — 4ac) 
SL  square,  we  cannot  find  the  required  values  of  x,  unless  we  can 
find  a  square,  which  subtracted  from  the  expression,  will  leave  a 
remainder  divisible  into  rational  factors.     But  in  this  case,  4ac  is 


190  *  .      ROBINSON'S   SEQUEL. 

greater  than  b^,  we  must  therefore  subtract  such  a  square  as  to 
diminish  a  and  c,  and  increase  6. 

To  accomplish  this  object,  we  will  subtract  the  square  of  (a? — 1), 
and  not  the  square  of  (x-\-\.) 

That  is,  from  13x^+15x-{-7y 

Subtract  a;^—  2x-{-\. 

Difference,  12x''-{-\7x-{-e. 

In  this  last  expression,  a=\2,  5=17,  and  c=^6.  ^ 

Hence,  b'^-^4ac=2S9—2QS=l,  a  square. 

We  are  now  sure  the  difference  is  divisible  into  rational  factors, 
and  to  obtain  the  factors,  we  put 

\2x^-\-\lx-{-Q=0. 

A  solution  of  the  quadratic,  gives  x= — f ,  or  x= — f . 

Whence,  (3i;-|-2)=0,  and  (4.2;-}-3)=0,  and  our  original  ex- 
pression becomes 

(^_l)3_^(3ar+2)  (4a?+3.) 

It  is  obvious  that  [x — 1 )  must  be  in  the  root,  and  one  of  the 
factors  may  be  in  tlie  other  branch  of  the  root  ;  that  is,  put 

(a;— 1)2_{-(3^+2)  (4;r+3)  =  [  [x—\)-\-n{3x-\-2)  y. 

By  reduction,         4x-\-3=:2n{x—\)-\-n^ (3x^2,) 

Or.  x=^"+3--^-. 

Take  w=l,  then  x=3,  and  \3x^ -\-\ 5X'\-1  ^\Q^ ,  a  square. 

(3.)  Find  such  a  value  of  x  as  will  render  14x^-|-5x — 39,  a 
square. 

After  a  few  trials  this  expression  is  found  to  be  the  same  as 
(2,r— l)2+(5.c— 8)  (2x-\-b.)  Assuming  its  root  to  be  2a:— 1+ 
p(5x — 8),  then  by  squaring  the  root,  making  it  equal  to  its  power, 

and  reducinor,  we  find  x=-^-^^^~^  . 
5p^-\-ip—2 

Assuming  p=  1 ,  x=  y ,  and  the  expression  equals  36,  a  square. 

Other  values  can  be  found,  by  assuming  different  values  to  jo. 

(4.)  Find  such  a  value  of  x  as  will  make  2x2-f"2^^~l~28,  <? 
square. 


ALGEBRA.  a  jgj 

After  a  little  inspection,  we  find  this  expression  equal  to 
(a:-|-4)2-f-(a;+l)  (x-\-l2.)     Now  if  we  make 

After  reduction,  we  shall  find  x= — Jl_"    ^   . 

J92— 2/>— 1 

Assume  ^=4,  then  x=4,  and  the  original  expression  is  144, 

a  square. 

If  (x-\-4y-{-(x+l)  (x+n)  =  [  (x+4)—p(x-\-12)y,  we  shall 

find  x=: ±- ±- If  we  take  p=l,  x=%.    If  we  take  »=| 

then  a;=8,  and  we  might  find  many  other  values  of  x  that  would 
answer  the  required  condition. 

Case  6th.  Expressions  in  the  form  a^x'^-\-hx^-\-cx^-\-dx-\'€y 
can  be  rendered  square,  provided  we  can  extract  three  terms  of 
their  square  roots. 

Assume  such  terms  as  the  whole  root,  making  its  square  equal 
to  the  given  expression,  and  the  resulting  value  of  x  will  make 
the  whole  expression  a  square. 

EXAMPLE. 

Find  such  a  value  of  x  as  will  make  4x''-|-4x^-(-4x^-|"2^ — ^'^ 
square. 

We  commence  by  extracting  the  square  root  as  far  as  three 
terms,  and  find  them  to  be  (2a;^-|-ar-|-|.) 

Therefore,         4x^  ^4x'^  ■\-4x^ -\-'lLx—%^{'ix'' -^-x-^-lY . 

Expanding  and  reducing,     2a; — 6=  |4:-[-t6- 
And  '        ;r=13|. 

Essentially  the  same  method  must  be  performed  in  other  ex- 
amples under  this  form.  ^ 

Case  7th.  Find  such  a  value  of  x  as  will  make  ax^-\'C,  a 
square. 

Expressions  in  this  form,  where  /;=0,  and  where  neither  a  nor 
c  are  squares,  nor  {h^ — 4ac)  a  square,  present  impossible  cases  : 
unless  we  can  first  find  by  inspection,  some  simple  value  of  x 
that  will  answer  the  condition. 

m 


Idt  «        ROBIKSON'S  SEQUEL. 

EXAMPLE. 

Find  such  a  value  of  x  as  will  make   2x^  -|-2,  a  square. 

It  is  obvious  that  if  x=\,  the  expression  is  a  square.  Now, 
having  found  that  1  will  make  the  expression  a  square,  we  can 
find  other  values  as  follows  : 

Leta:=l-[-y;  then  x^  =^\-\-2y-\-y^ , 

And         2x''+2=4+4y-\-2yK 

Here  the  original  expression  is  transformed  into  another  expres- 
sion, having  a  square  for  its  first  term. 

Now  we  must  find  such  a  value  of  y  as  shall  make  4-|-4y-|-2y^, 
a  square. 

Assume  4-[-42/-|-22/"=(2 — my)^=4 — 4my-\^m^y^.       Or, 

4-|-2y= — 4m4-m^y.     Hence,  y=-^ — —-,  ^  may  be  any  num- 

m^ — 2 

ber  greater  than  one.  Put  m=2.  Then  y=6,  and  a;=l-f-y=7, 
and  the  original  expression,  2a?2-|-2=98-|-2=:100,  a  square. 

N.  B.  It  often  occurs  incidentally  in  the  solution  of  problems, 
that  we  must  make  a  square  of  two  other  squares.  This  can  be 
done  thus:     Let  it  be  required  to  make  a:^-|-y^,  a  square. 

Assume  x=p^ — q^,  and  y=2j)q. 

Then  x''=2)^—2p^q^-\-q\ 

And  y^=  4p^q^, 

Add,  and  x^ -\-y^  ^^p'^ -\'2p^ q"^ -\-q^ y  which  is  evidently  a 
square,  whatever  be  the  values  of  p  and  q.  We  can,  therefore, 
assume  j9  and  q  at  pleasure,  provided^  be  greater  than  q. 


%  SECTION   X. 

DOUBLE  AND  TRIPLE  EQUALITIES. 

We  have  thus  far  confined  our  attention  to  finding  a  value  of  x 
that  would  render  a  single  expression  a  square.  Now  we  propose 
finding  a  value  of  x  that  will  render  several  expressions  squares 
at  the  same  time. 


ALGEBRA.  193 

Case  1st.  As  a  general  expression  for  double  equality,  let  it 
be  required  to  find  such  a  value  of  x,  that  will  make  (ax-\-b)  and 
(ca:-[-c?),  squares  at  the  same  time. 

X:=: . 

a 

^2 ^ 


Whence,  -^ =- ,  or  d^ — cb=ap^ — ad. 

a  c 

Transposing  cb,  and  multiplying  by  c,  gives 
c^t-=  acp  ^  -\-c  2  d — acd. 

As  the  first  member  of  this  equation  is  square  for  all  values 
of  c  and  t,  it  is  only  requisite  to  find  such  a  value  of  p,  as  to 
make  the  second  member  a  square  ;  which  can  be  done  by  some 
of  the  artifices  heretofore  explained. 

To  illustrate,  we  give  the  following  definite  problem  : 

The  double  of  a  certain  number  increased  by  4,  makes  a  square ; 

and  Jive  times  the  same  number  increased  by.  1,  makes  a  square. 

What   is  that  number? 
Let  X  be  the  number  ;  then 


2ar+4=^2 


Whience,  \  ^ 


Then  5t''—^0=2p''—^. 

And  25/2  =  10^2+90. 

The  first  member  of  this  equation  is  a  square,  whatever  be  the 
value  of  f ;  and  all  the  conditions  will  be  satisfied,  provided  we  can 
find  such  a  value  of  p  as  to  make  the  second  member  a  square. 

This  expression  corresponds  to  case  7,  and  we  cannot  proceed, 
unless  we  find  by  trial,  by  intuition  as  it  were,  some  simple  value 
of  ^  that  will  make  10p2_|_9Q^  a,  square;  and  we  do  perceive 
that  jt?=l  will  make  the  whole  expression  100,  a  square. 

Now  if  jt>=l  will  give  a  definite  and  positive  value  to  ar,  which 
will  answer  the  required  conditions,  the  problem   is  solved.     If 
not,  we  must  find  other  values  of  p. 
13 


sions 


fH  ROBINSON'S  SEQUEL. 

Here  x=^ /and  i(p=l,  x=0,  and  the  expressions,  2a:-|-4 

6 

and  5a:-|-I,  become  4  and  1.     Squares,  it  is  true,  which  answer 

the  technicalities,  but  not  the  spirit  of  the  problem. 

To  find  another  value  of  jt?,  put  jO=  !-[-?•     Then 

1 0^2  _^90=  1 00+20jt>+2^^ 

To  make  this  a  square,  assume 

\00-{-20q-{-2q''  =  (l0— nqy  =  100— ^Onq+n^g'*. 

By  reduction,  q= — l_ir_''.    Now  n  must  be  so  taken,  that  n' 
n^ — 10 

will  be  greater  than  10;  take  w=5  and  §'=8,  p=Q,  then  x=l6, 

and  the  original  expressions,  2^-j-4=36,  a  square,  and  5x-\-l=z 

81,  a  square. 

Case  2d.     A  double  equality  in  the  forpi  ox'^-\-bx=Ci*  and 

cx^-\-dx=0,  may  be  resolved  by  making  a:=--,  then  the  expres- 

y 

will  become -^(a-j~*y)  ^^^ — (^+^y)»  which  must  be 

made  squares. 

But  if  we  multiply  a  square  hy  a  square^  or  divide  a  square  hy 
a  square,  the  product  or  quotient  will  he  square. 

Now  as  each  of  the  preceding  expressions  are  to  be  squares, 

and  as  they  obviously  have  a  square  factor — ,  it  is  only  necessary 

y^ 

to  make  a-\-by,  and  c-\-dy,  squares,  as  in  the  first  case. 

We  may  also  take  another  course  and  assume  ox^-\-bx=p^x^ , 

which  gives  x= ,  whic^h  value  put  in  the  other  expression, 

p^ — a 

and  we  have  c( )  -\-d( )  =  D . 

\p^—a/     '     \p^—a/ 

Multiplying  this  by  the  square  [p^ — aY ,  and  the  expression 
becomes  ch^ — dhd-\-abp^=  some  square,  from  which  the  value 
of  p  can  be  found,  and  afterwards  x. 

EXAMPLE. 

Find  a  numher  whose  square  increased  by  the  number  itself,  and 
*Thi8  symbol  is  read  a  square. 


ALGEBRA.  196 

whose  square  diminished  by  the  number  itself,  the  sum  and  difference 
shall  be  squares. 

Let  ar=  the  number  ;  then  by  the  conditions, 

a;2-|-a:=n,  and  x^ — a;=  some  other  square. 


Assume  x=  -  ;  then  — (  1  -f-y  )  =  D . 


The  first  members  of  these  equations  are  obviously  squares, 
provided  the  factors  (l-(-y)  and  (1 — y)  are  squares. 

To  make  these  factors  squares,  put 

l-|-y=jt?2,  and  1 — y=^q^  • 

Whence,  y==p^ — l,andt/=l — g*. 

p^z=<2.—q^. 

All  the  conditions  will  be  satisfied,  when  we  discover  what 
value  must  be  given  to  q  to  make  (2 — q^),  a  square  ;  and  q=\ 
satisfies  that  condition. 

This  value  oi  q  makes  j9=l,  and  y=0. 

But  ic=:_=_=  infinity. 

y     0 
If  X  is  infinite,  x^  can  neither  be  increased  or  diminished  by 
adding  and  subtractings;;  ihereioxQ  x^-\-x=i\2,  vm^  x^ — ^•=n, 
because  x^  is  obviously  a  square. 

But  practically,  we  say  that  this  value  of  q  will  not  answer  the 
conditions  ;  therefore,  we  will  find  other  values  as  follows  : 
Put         q=\-\-t. 
Then  2— y^  ^i__2t—i'' =^(\~ntY  =  \—ZrU+n'' f" . 

Or,  /=?l!^i.     Take  n=2, 

n^+\ 
Then^=|.   ^=1+1=1.     y=i_||===_|4. 

1  25 

But  x=-= — --  ,  the  number  souofht. 

y       24' 

Those  who  desire  a  positive  number,  can  take  n  negative. 

Case  3d.     To  resolve  a  triple  equality. 

Equations  in  the  form  cx-\-hy=:it'^ ,  ax-\-dy—u^,  eX'\-fy='8^ , 
can  be  resolved  thus  : 


J96  BOBINSON'S   SEQUEL. 


By  eliminating  x,  we  find  y= 
By  eliminating  y,  we  find  x 


au 


ad — be 

ad — be 

Substituting  these  values  of  a;  and  y  in  the  equation  &c-[^y=5^, 
we  shall  have 

(af-be)u-+{de-ef)e_^,  . 
ad — be 
Assume  e«=r±:^2;;  then  u^=^t^z^,  and  put  this  value  in  the 
above,  and  divide  by  t^ ,  then 

{af—bc)z^J^[de—cf)_  s2 
ad— be  '¥' 

As  the  second  member  of  this  equation  is  a  perfect  square,  all 
the  conditions  will  be  satisfied  when  we  find  a  value  of  z  that 
will  render  the  first  member  a  square.  This,  when  possible^  can 
be  done  by  case  7,  of  section  viii. 

After  z  is  found,  t  can  be  assumed  of  any  covenient  value 
whatever.     When  u  and  i  are  known,  x  and  y  will  be  known. 

We  are  now  through  with  theory, — not  that  we  have  presented 
the  whole,  there  are  some  cases  in  practice  that  no  general  rules 
will  meet,  and  the  operator  must  depend  on  his  own  judgment 
and  penetration. 

Much,  very  much,  will  depend  on  the  skill  and  foresight  displayed 
at  the  commencement  of  a  problem,  by  assuming  convenient  ex- 
pressions to  satisfy  one  or  two  conditions  at  once,  and  the  remain- 
ing conditions  can  be  satisfied  by  some  one  of  the  preceding  rules. 

EXAMPLES. 

(1.)  It  is  required  to  find  three  nwmbers  in  arithmetical  progres- 
sion,  such  that  the  sum  of  every  two  of  them  may  be  a  square. 

Let  X,    x-\-y,  and  ar-|-2y,  represent  the  numbers. 
Then  by  the  general  formula, 

^x^y=t^,  2x-\-2y=:u^,  2x-\-Sy—s^, 

^  _  ■  /2 y  21  2  2,11 

By  extermmating  x,  we  have -= ^  . 


ALGEBRA.  197 

Continuing  thus  according  to  the  general  equations,  we  must 
go  through  a  long  and  troublesome  process,  and  in  conclusion  we 
shall  find  the  numbers  to  be  482,  3362,  and  6242. 

Another  Sdviion. 

Observing  the  remark  immediately  preceding  the  example,  we 

put  — — y,  — ,  and  — -|-y  to  represent  the  numbers. 

Ai  ^  At 

Then  {x^ — y),  (^^+y),  and  x^  must  be  the  squares.  But  x^ 
is  a  square  for  all  values  of  x ;  therefore,  we  have  only  to  make 
squares  of  (x^ — y)  and  {x'^-^-y-) 

Let  y=2x-\-l  ;  then  x^ -\-y=x'^ -\-2x-\-l ,  a  square  for  all  values 
of  X.  Hence,  all  we  have  to  do  is  to  find  a  value  of  x  that  will 
make  a  square  of  the  expression  x^ — y,  or  x^ — 2x — 1.  Assume 
the  square  root  to  be  (x — n)  ;  then 

x^  — 2x —  l=x'^  — 2nx-\-n^  • 

x=.^L+l_ 
2(n—l) 

Take  w=ll,  then  a;=:6.l,  and  -1^:2  =  18.605,  y=13.2,  and  this 
numbers  are  5.405,  18.605,  and  31.805.  Various  other  numbers 
may  be  found,  by  giving  different  values  to  n. 

(2.)  Find  two  numbers  such  that  if  to  each,  as  also  to  their  sum^ 
a  given  square  el^  be  added,  the  three  sums  shall  all  be  squares. 

Let  x^ — a^,  and  y- — a^  represent  the  numbers  ;  then  the  first 
conditions  are  satisfied. 

It  now  remains  to  make  x^-{-y^ — 2a^-\-a^  a  square,  or,  x^-\' 
y 2 — Qj2 __  [-]  ^  Assume  y^  — a^  ==:2ax-\-a ^ .  This  assumption  will 
make  the  expression  a  square,  whatever  be  the  values  of  either 
a;  or  a.  But  the  assumed  equation  gives  y^  =^2ax-\-2a^ ,  a.nd  as 
y^  is  a  square,  we  must  find  such  values  of  x  and  a,  as  shall  make 
2aa:-|-2a2,  a  square.  Put  x=-na.  Then  2wa2-|-2a^  =  n,  or, 
a^  (2w+2)=  D .  Hence  it  is  sufficient  that  we  put  2w-|-2=  some 
square.  Therefore,  assume  2w-|-2=l6.  Hence,  w=7  and  ir=7a. 
Now  take  a  equal  to  any  number  whatever.  If  a=l,  a:=7,  y=4, 
and  48  and  15  are  the  numbers,  add  1  to  each,  and  we  have  49 
and  16,  squares  ;  sum,  63-|-l=64,  a  square. 


198  ROBINSON'S  SEQUEL. 

(3.)     Find  three  square  nunilers  whose  sum  shall  he  a  square. 

Letir*+y2_f-22_Q^  Assume  y'^='^z.  Then  xf -{-2xz-^z^ 
is  a  square.  But  2a:s=D.  Let  x=uz,  then  ^uz^  =  {j,  or  2u= 
□  =  16,  u=S,  x=Qz,  z—\,  x—^,  y=4. 

Therefore  64+16+1=81=92. 

(4.)     Fhui  three  square  numbers  in  arithmetical  progression. 

Let  x^ — y,  x^,  and  x^-\-y  represent  the  numbers.  Assume 
a.2__y2_|_i^  then  the  first  and  last  will  be  squares,  and  it  only  re- 
mains to  make  (y^+i),  a  square. 

Therefore,  put  3/2+1  =(y—j9) 2.     Whence,  y=£- !. 

2p 
Take^=l,  then  2/=|,  and  y2+|=f|=a;2. 

Consequently,  -g^,  ||,  and  ^\  are  the  numbers  ;  but  we  can 
multiply  them  all  by  the  same  square  number,  64,  without  chang- 
ing their  arithmetical  relation^  and  their  products  will  still  be 
squares,  1,  25,  and  49.  Multiplying  these  numbers  by  any  square 
number,  will  give  other  numbers  that  will  answer  the  condition. 

(5.)  Find  two  whole  numbers,  such  that  the  sum  and  difference  of 
their  squares  when  diminished  by  unity,  shall  be  a  square. 
Let  a:+l=:  one  number,  and  y=  the  other. 
Then  by  the  conditions,     x'^-\-y^-\-^x=.{2  (1) 

And  a:2— y2+2ar=n  (2) 

Assume  2a:=a2,  and  3/^=2cu;;  then  (1)  and  (2)  become 
(a;2+2aa;+a2)  and  (a:^— 2aa;+a2), 

obvious  squares  whatever  may  be  the  values  of  x  and  a. 

But  the  equations  9,x=ia^ ,  and  y^  =2ax,  must  be  satisfied. 
Take  a=4,  then  x=S,  a;+l=9,  and  9  and  8  are  the  numbers 
required. 

(6.)  FtTid  three  whole  numbers,  such  that  if  to  the  squares  of  each, 
the  product  of  the  other  two  be  added,  the  three  sums  shall  be  square*. 
Let  a?,  xy,  and  xv  be  the  numbers. 
Then  by  the  conditions,     x^  -\-x^vy=  □ . 
x^y^-{-x'^v=0. 
And  x^v^^x^yz=[2. 


ALGEBRA.  199 

Omitting  the  common  square  factor  x^,  it  will  be  sufficient  to 
make  squares  of  the  following  expressions  : 

\-\-vy=U. 

Assuming  y=4v-\-4  will  make  the  first  and  third  expressions 
square. 

Substituting  the  value  of  y^  in  the  second  expression,  we  shall 
have  16y^-f-33y-j-16,  which  must  be  made  a  square. 

Whence,  \ev^+^3v-\-16={4—pvy. 

Reduced,  gives  v= — X-^.     Take  ^=5. 

p'^ — 16 

Then  v=\^.  Now  take  x=9,  and  9,  73,  328,  will  be  the  re- 
quired numbers. 

(7.)  Find  two  whole  numbers  whose  sum  shall  be  an  integral  cube, 
and  the  sum  of  their  squares  increased  by  thrice  their  sum,  shall  he  an 
integral  square. 

Let  x-^y=^n^,  that  is,  some  cube.  Then  x^-^y^-\-Zn^=z\2' 
Put  2;ry=37i3,  then  x^  -\-9,xy-^y^  is  a  square,  whatever  may  be  the 
values  of  x  and  y.  But  x  and  y  must  conform  to  the  equations 
x-\-y,=n^,  and  2xy=Sn^.  Work  out  the  value  of  x  from  thesQ 
equations,  on  the  supposition  that  n  is  known,  and  we  shall  find 
2x=n^-{-J{n^—6n^}. 

Now  a;  will  be  rational,  provided  we  can  find  such  a  value  of  n 
as  shall  render  n^ — 6n^  a  square,  but  if  we  add  9  to  this,  we 
perceive  it  must  be  a  square,  and  we  have  two  squares,  which 
difi"er  by  9.  Therefore  one  must  be  16,  the  other  25,  as  these 
are  the  only  two  integral  squares  which  differ  by  9.  Hence, 
;i6_6^3_|_9^25.  Or,  ^3—3=5.  n^  =  ^,  n=2,  and  x=6, 
y=2. 

(8.)  Find  three  numbers  such  that  their  sums,  and  also  the  sum 
of  every  two  of  them,  may  all  he  squares. 

Let  x^ — Ax=  the  first,  4x=  second,  and  2:r-}-l=  third.  By 
this  notation,  all  the  conditions  will  be  satisfied,  except  the  sum  of 
the  last  two.  That  is  62"-|-l  must  be  a  square,  but  to  have  three 
different  whole  numbers,  no  square  will  answer  under  1 2 1  ,.the  square 


200  ROBINSON'S   SEQUEL. 

of  11.     Hence,  put  6a;-[-l  =  121.     Or,  a;=20.     And  the  numbers 
will  be  320,  80,  and  41. 

(9.)  Find  two  numbers  such  thai  their  difference  may  he  equal  to 
the  difference  of  their  squares,  and  the  sum  of  their  squares  shall  bf 
a  square  number. 

Let  X  and  y  be  the  numbers.  Then  x — yz=x^ — y^ .  Divide  by 
X — y,  and  l=x-\-y.  Hence  a?=l — y,  and  x^-\-y^  =  \ — 22/-(-2y^. 
Which  last  expression,  1 — 2y-|-2y^,  must  be  made  a  square.    For 

this  purpose  put  1—22/4-2^2 =(1 — nyY .     Hence,  y=5i^IIl  i. 

n^ — 2 

Take  n  any  value  to  render  y  less  than  one,  in  order  to  make 

X  positive.     Take  w=3,  then  y=y,  and  a;=f ,  the  answer. 

The  following  are  not  difficult,  and  we  leave  them  as  a  pleas- 
ant exercise  for  learners. 

(10.)  Find  three  numbers  in  geometrical  progression,  such  that  if 
the  rman  be  added  to  each  of  the  extremes,  the  sums  in  both  cases  shall 
be  squares.  Ans.  6,  20,  and  80. 

(11.)  Find  three  numbers,  such  that  their  product  increased  by 
unity  shall  be  a  square,  also  the  product  of  any  two  increased  by  unity, 
shall  be  a  square.  Ans.  1,  3,  and  8. 

Assume  1  for  the  first  number,  and  x  and  y  for  the  other  two. 

(12.)  Find  two  numbers,  such  that  if  the  square  of  each  be  added 
to  their  product,  the  sums  shall  be  both  squares.      Ans.  9  and  16. 

(13.)  Find  three  integral  square  numbers  in  harmonical  propor- 
tion. Ans.  25,  49,  and  1235. 

(14.)  Find  two  numbers  in  the  proportion  of  S  to  15,  and  such 
that  the  sum  of  their  squares  shall  be  a  square  number. 

Ans.  156  and  255.     Bonnycastle's  answer  is  476  and  1080. 

(15.)  Find  two  numbers  such  that  if  each  of  them  be  added  to  their 
product,  the  sums  shall  be  both  square.  Ans.  ^  and  |. 

We  have  given  as  much  on  this  topic  as  will  be  profitable,  save 
the  following  remote  and  partial  application. 


ALGEBRA.  201 

EXAMPLES. 

(1.)    Given  -j  ^2_  — 7  [  ^^  ^^^  *^®  values  of  x  and  y. 

x"  ={l—y.)  y2^(74-a;.)  Here  (7— y)  and  {1+x),  must  be 
squares.  x=2,  and  y=3,  will  evidently  answer  the  conditions  ; 
and  as  these  values  will  verify  the  given  equations,  the  solution 
is  accomplished. 

(2.)     Given  |    l^^Zu'^y^'TLixy'^d  \  ^^  ^°^  ^^^^^^  ^^  ^  ^^^  ^' 
As  4  and  9  are  squares,  the  first  members  are  square  in  fact, 
though  not  in  form.     But  we  can  make  the  first  members  square 
in  form,  by  assuming 

2.c2 — 2,xy—0,  and  Sy^— 2.ry=0. 
Then  y^=4  and  a;2=9,  or  y=2  and  a;=3  ;  values  which  ver- 
ify all  the  equations, 

(3.)    Find  such  integral  values  of  x,  y,  and  z,  as  will  verify  the 

equations 

x^+y^+xy=^l. 

And  a;2-j-s^+^2=49. 

If  we  add  xy  to  the  first  equation,  and  xz  to  the  second,  the 
first  members  will  be  square  ;  and,  of  course,  the  second  mem- 
bers will  be  square  in  fact,  though  not  in  form. 

We  have  then  to  make  ^l-\-xy,  and  ^^-\-xz,  squares. 

To  accomplish  this,  put  37-|-a^=49,  or  xy=\^         (1) 

And  49+^2=64,  or  ii:s= 15         (2) 

From  (1),      2:=-;   from  (2),  a:=— . 
y  z 

Hence,      122=15?/,  or  2=—. 

Take  y=4,  then  2=5,  and  a;=3  ;  values  which  will  verify  the 
given  equations. 

(4.)  Find  such  integral  values  of  y  and  z  that  will  verify  the 
equation  y'^ -\-z^ -\-yz=Q\ . 

Add  yz  to  both  members,  then  put   Q\-\-ys=^n^ . 

Now  if  we  assume      w=8,     yz=^2>. 

But  yz=i'^  will  give  y-|-2=8,  and  these  two  equations  will  not 
give  integral  values  to  y  and  z.  Therefore,  take  ?z=9,  then 
w=^=81,  y2=20,  y-|-^=9.     Hence,  2=4  or  5,  and  y=5  or  4. 


202  ROBINSON'S  SEQUEL. 

(5.)    Given  ■!  a.  ^ajx^ZL  r  f  *^  ^°^  ^^  values  of  x  and  y. 

Put      xy=^py  transpose,   &c. 

Then  4a;2  =  12+2/>.     Ay^  =  \Q—Qp. 

Now  if  we  find  such  a  value  of  j9  as  will  make  (IS-f-Sp)  and 
(16 — Qp),  squares  at  the  same  time,  it  is  highly  probable  that  such 
a  value  will  verify  the  original  equations.  It  is  obvious  that  ^=2, 
will  make  the  expressions  squares ;  then  4a:^  =  16,  a;=2  and  2/=l, 
and  these  values  will  verify  all  the  equations. 


,^\     n-  \  6a;24-2v^  =5^4-12  )     to  fin( 

(6.)    Given  j  3^,:[:2^^_3^?1  3  ^    ^^^^^ 


d  one  value  of  x 


This  problem  is  under  (Art.   1 10,  alg.) 

Add  the  equations  together,  and  reduce,  and  we  have 
9.r2=y2_|_3a:y+9. 

The  first  member  of  this  equation  is  a  square  ;  therefore  the 
second  member  is  a  square,  but  to  make  it  a  square  in  form,  as 
well  as  in  fact,  we  perceive  it  is  only  necessary  to  make  a;=2. 
Then  dx^  =:y^ -^-Qy-^-^ ,  and  3.r=y-|-3  ;  whence  y=3,  and  these 
values  verify  the  given  equations. 

This  method  of  operation  must  be  used  with  great  caution,  and 
taken  for  just  what  it  is  worth. 

2  2 

(7.)  Given  ar-|-y=35,  and  x^ — y^^=5,  to  find  the  values  of  x 
and  y. 

Put     x^=F,  and   y^=Q. 

Then  P^+Q^=35,  and  F^—Q^=5. 

Or,  P^=35—Q\  and  F^=5+QK 

The  equations  can  all  be  verified,  provided  we  find  can  such  a  val- 
ue of  Q  that  will  make  (35 — Q^)  a  cube,  and  (5-\-Q^),  a  square. 

We  will  try  the  next  less  integral  cube  below  35.  That  is,  we 
will  assume    35—^3^27.      Then    Q=2,   and  (5-{-Q^)=9,   a 

square.     Then  P=3,  and  x^==3,  or  a;=27,  and  y=8. 

This  problem  was  given  in  the  first  editions  of  Robinson's 
Algebra,  page  147,  under  the  head  of  pure  equations,  but  it  was 
out  of  place  and  is  now  changed. 


PART  THIRD 


SECTION   I. 


OEOMrETRY. 


D  right 


Thirty-one  of  the  following  problems  will  be  found  in  Robinson's 
Geometry,  commencing  on  page  100. 

(1.)  From  two  given  points,  draw  two  equal  straight  lines  which 
shall  meet  in  the  same  point  in  a  line  given  in  position. 

Let  A  and  B  be  the  two  given 
points,  taken  at  pleasure,  and  MO 
the  line  given  in  position. 

Join  AB  and  bisect  it  in  J). 
Draw  J)U  perpendicular  to  AB,  to 
meet  the  line  IIO  in  U.  Join  AH 
and  BJEJ,  the  lines  required.  Be- 
cause AJ)=^DB,  and  DE  com- 
mon to  the  two  A's  ADE,  BDE 
angles,  therefore  AE=^BE.     Q.  E, 

N.  B.  For  simple  and  obvious  demonstrations,  we  shall  not  go 
through  the  steps  in  full,  but  refer  to  Robinson's  Geometry  for 
the  proposition  that  applies. 

(2.)  From  two  given  points  on  the  same  side  of  a  line  given  in 
position,  to  draw  two  lines  which  shall  meet  in  that  line  and  make 
equal  angles  with  it. 

Let  A  and  B  be  the  two  given  points, 
and  HO  the  line  given  in  position. 

From  one  of  the  given  points  as  B, 
let  fall  the  perpendicular  B  0,  to  the 
given  line,  and  produce  it  to  D,  making 
0D=B0. 

Then  join  AD  :  this  line  will  neces- 
sarily cut  the  hne  HO  in  some  point  E. 
Join  EB,  and  AE  and  EB  are  the  re- 
quired lines.     jLBEO=I^DEO,  (Book  1,  Th.  13.)     L.AEH= 
L,DEO,  (Th.3,  Bookl.)   Whence,  L.BE  0=1^  A  EH.  Q.E.D. 

203 


204 


ROBINSON'S  SEQUEL. 


(3.)  If  from  any  jiolnt  without  a  circle,  two  straight  lines  he  dravm 
to  the  concave  part  of  the  circumference,  making  equal  angles  with  the 
line  joining  the  same  point  and  the  center ;  the  parts  of  these  lines 
which  are  intercejjted  loithin  the  circle,  are  equal. 

Let  A  be  the  point  without 
the  circle.  Join  A  C  and  draw 
any  other  hne  to  cut  the  ciroie 
as  AD  ;  then  draw  AB  so  that 
the  angle  CAB=^  CAD.  Then 
we  are  to  show  that  FB^ED. 

The  two  A's,  ABC  and 
ADC,  having  two  sides  AC, 
CB,  of  the  one,  equal  to  A  C, 
CD,  of  the  other,  and  their  re- 
spective angles  at  A  equal,  the 
two  A's  are  equal.  That  is, 
AB=AD.  For  the  same  rea- 
son the  two  A's  ACF,  ACE 
are  equal,  and  AF=AE. 

Whence,   An—AF=zAD~AE,  or  BF=DE.     Q.  E.  D. 


(4.)  If  a  circle  be  described  on  the  radius  of  another  circle,  any 
straight  line  drawn  from  the  point  where  they  meet,  to  the  outer  cir- 
cumference, is  bisected  by  the  interior  one. 

Let  A  C  be  the  radius 
of  one  circle  and  the  di- 
ameter of  another,  as 
represented  in  the  figure. 
From  the  pointof  contact 
A,  of  the  two  circles, 
draw  any  line,  as  All; 
this  line  is  bisected  in  D. 
Join  DC  and  ffB.  Then 
ADChemg  in  a  semicircle,  is  a  right  angle;  also,  AJIB  is  a 
right  angle,  for  the  same  reason:  therefore,  DC  and  HB  are 
parallel.     Whence, 

AD    :    Aff    :    :    AC    :    AB 


GEOMETRY. 


205 


But  as  AB  is  double  of  A  C,  therefore  All  is  double  of  AD, 
or  ^^is  bisected  in  I).     Q.  E.  D. 


(5.)  JF'rom  two  given  points  on  the  sante  side  of  a  line  given  in 
position,  to  draw  two  straight  lin£S  which  shall  contain  a  given  angle, 
and  be  terminated  in  that  line. 

Let  A  and  B  be  the 
two  given  points  and  j0"(9 
the  line  given  in  posi- 
tion. For  the  sake  of 
perspicuity,  we  will  re- 
quire two  lines  drawn 
from  the  two  points,  A 
and  B,  to  meet  in  HO, 
and  make  an  angle  of 
50°.  Subtract  50  from 
1 80,  and  divide  the  re- 
mainder by  2,  this  pro- 
duces 65°. 

At  A  make  the  angle 
BAC=Qb°,  and  at  B 
make  the  angle  ABC=^Qb°  ;  these  two  lines  will  meet  in  C, 
making  an  angle  of  50°.  About  the  A  ABC  describe  a  circle, 
cutting  HO  in  //and  0.  Join  AH,  BH  AHB  is  equal  ACB, 
(th.  9,  b.  iii,  scholium,)  the  angle  required. 

Lines  drawn  from  A  and  B,  to  meet  the  line  in  0,  would  also 
answer  the  conditions, 

N.  B.  When  the  given  angle  is  not  sufficiently  small  to  cause 
the  angle  C  to  fall  below  the  line  HO,  the  problem  is  impossible. 

(6.)  If  from  amj  point  without  a  circle,  lines  he  drawn  touching  it, 
the  angle  contained  hy  the  tangents  is  double  of  the  angle  contained  by 
the  line  joining  the  points  of  contact,  and  the  diameter  drawn  through 
one  of  them. 

This  problem  requires  no  figure.  Imagine  a  point  without  a 
circle,  a  line  drawn  from  that  point  to  the  center  of  the  circle, 
and  lines  drawn  to  touch  the  circle  on  each  side.     Join  the  points 


206  ROBINSON'S  SEQUEL. 

of  contact  and  the  center  of  the  circle.  Thus  we  have  two  equal 
right  angled  triangles,  having  the  same  hypotenuse,  the  line  from 
the  given  point  without  the  circle  to  the  center  of  the  circle. 
With  the  correct  figure  in  the  mind,  the  truth  of  the  proposition 
is  obvious. 


(7.)  If  from  any  two  points  in  the  circumference  of  a  circle,  there 
he  drawn  two  straight  lines  to  a  point  in  a  tangent  to  thai  circle,  they 
ivillmake  the  greatest  angle  when  drawn  to  the  point  of  contact. 

Let  A  and  B  be 
the  two  points  in  the 
circle,  and  CD  a  tan- 
gent line.  The  prop- 
osition requires  us  to 
demonstrate  that  the 
angle  A  GB  is  greater 
than  the  angle  ADB. 
ACB=AOB, {th.9, 
b.  iii,  sch.)  But  A  OB  is  greater  than  ADB,  (th.  11,  b.  i,  cor. 
1),  therefore,  ACB  is  also  greater  than  ADB.     Q.  E.  D 

(8.)  Fro?n  a  given  point  within  a  given  circle,  to  draw  a  straight 
line  which  shall  make  with  the  circumference  an  angle  less  than  any 
angle  made  by  any  other  line  drawn  from  that  point. 

Let  P  be  the  given 
point  within  the  circle, 
and  C  the  center.  Join 
PC.  Through  P  draw 
APB  at  right  angles  to 
PC.  Also,  through  P 
draw  any  other  line  as 
P  G  ;  then  we  are  to  show 
that  PBt  is  less  than 
PGH. 

From  C  let  fall  the 
perpendicular  CD  on  the 
chord  FG.  PC  is  the 
hypotenuse  of  the  right 


'4i 


GEOMETRY.  207 

angled  triangle  PDC \  therefore,  PC  is  greater  than  CD,  con- 
sequently the  chord  FO  is  greater  than  the  chord  AB,  (th.  3, 
b.  iii.)  and  the  arc  OAF  is  greater  than  the  arc  BOA.  The 
angle  POHis,  measured  by  half  the  arc  OAF,  and  PBt  is  meas- 
ured by  half  the  arc  BOA  ;  therefore,  the  angle  POH  is  greater 
than  the  angle  PBt,  or  PBt  is  less  than  POH.     Q.  E.  D. 

N.  B.  The  angle  which  any  chord  makes  with  the  circumfer- 
ence, is  the  same  as  between  the  chord  and  tangent, — because  the 
circumference  and  tangent  unite  as  they  meet  the  chord. 

(9.)  If  two  circles  cut  each  other,  the  grectiest  line  that  can  he 
draion  through  the  point  of  intersection,  is  that  which  is  parallel  to  the 
line  joining  their  centers. 


Let  A  and  B  be  the  center  of  two  circles  which  intersect  in  0. 
Through  0  draw  mn  inclined  to  AB, — then  we  are  to  prove  that 
mn  is  less  than  it  would  be  if  it  were  parallel  to  AB.  Draw  AC 
and  BI)  perpendicular  to  mn,  then  CD^=\mn.  Draw  (7^ paral- 
lel to  AB,  then  CH=AB  ;  and  CZT being  the  hypotenuse  of  the 
right  angled  A  CDH,  GH,  or  its  equal  AB,  is  greater  than  CB. 
Now  conceive  mn  to  revolve  on  the  center  0,  until  CD  becomes 
parallel  to  AB  ;  CD  will  then  become  equal  to  AB.  But  mn 
will  be  all  the  while  double  of  CD  :  therefore,  mn  will  be  the 
greatest  when  parallel  to  AB.     Q.  E.  D. 

(10.)  Iffrofrti.  any  point  within  an  equilateral  triangle,  perpendic- 
ulars he  drawn  to  the  sides,  they  are  together,  equal  to  a  perpendicular 
drawn  from  any  of  the  angles  to  the  opp)osite  side. 


208 


ROBINSON'S  SEQUEL. 


Let  ABC  be  the  equilateral  A, 
CD  a  perpendicular  from  one  of  the 
angles  on  the  oposite  side ;  then  the 
area  of  the  A  is  expressed  by  \AB 
X  CD.  Let  P  be  any  point  within 
the  triangle,  and  from  it  let  drop  the 
three  perpendiculars  FG,  PH,  P  Oy 

The  area  of  the  triangle  APB  is 
expressed  by  ^ABy^PG.  The  area 
of  the  A  CPB  is  expressed  by 
\CBxPO\  and  the  area  of  the  A 
CPA  is  expressed  by  ^CAy^PH.  By  adding  these  three  expres- 
sions together,  (observing  that  CB  and  CA  are  each  equal  to 
AB,)  we  have  for  the  area  of  the  whole  A  ACB,  \AB{PG+ 
PH-\-PO.) 

Therefore,         \  ABX  CD==^AB{PG+Pir+P  0.) 

Dividing  by  i^^,  gives    CD=PG-\-PH+P  0.     Q.  E.  D. 

(11.)  If  the  points ,  bisecting  the  sides  of  any  triangle  he  joiried, 
the  triangle  so  formed,  will  be  one-fourth  of  the  given  triangle. 

If  the  points  of  bisection  be  joined,  the  triangle  so  formed  will 
be  similar  to  the  given  A,  (th.  19,  b.  ii.) 

Then,  the  area  of  the  given  A  will  be  to  the  area  of  the  A 
formed  by  joining  the  bisecting  points,  as  the  square  of  a  line  is  to 
the  square  of  its  half ;  that  is,  2^  to  1,  or  as  4  to  1.  Hence  the 
A  cut  off  is  I  of  the  given  A-     Q.  E.  D. 

{\9..)  The  difference  of  the  angles  at  the  base  of  any  triangle,  is 
double  the  angle  contain£d  by  a  line  drawn  from  the  vertex  perpen- 
dirular  to  the  base,  and  another  bisecting  the  angle  at  the  vertex. 

Let  ^^C  be  a  A.  Draw  A3f  bi- 
secting the  vertical  angle,  and  draw 
AD  perpendicular  to  the  base. 

The  theorem  requires  us  to  prove  that 
the  diferenne  between  the  angles  B  and 
Cis  double  of  the  angle  MAD. 

By  hypothesis,  the  angle  CAM= 
MAB.     That  is,       CAM=MAD-\-DAB,  (1) 


GEOMETRY. 


209 


(  C+CAM+MAJ) 
By  (th.  11,  b.  i,  cor.  4.)   -j  j^j^j)^j^ 


:90°.  I       (2) 

:90°.  f        (3) 

Therefore,  B+DA£=C-{-CAM-\-MAD.        (4) 

Taking  the  value  of  CAM  irom.  (1),  and  substituting  it  in  (4), 
gives  B+DAB=  C-\-3fAI)+DAB+MAD. 

Reducing,  (B—C)=2MAD.         Q.  E.  D. 


(13.)  If  from  the  three  angles  of  a  triangle,  lines  be  drawn  to  the 
middle  of  the  op-ponte  sideSy  these  lines  will  intersect  each  other  in  the 
same  point. 

Let  ABC  be  a  A,  bisect 
BCmE,  AC'mF. 

Join  AU  and  BF,  and 
through  their  point  of  inter- 
section 0,  draw  the  line 
CD.  JVow  if  ice  prove  AD 
=DB,  the  theorem  is  true. 

Triangles  whose  bases  are 
in  the  same  line,  and  vertex  in  the  same  point,  are  to  one  another 
as  their  bases  ;  and  when  the  bases  are  equal,  the  triangles  are 
equal.  For  this  reason  the  A  AFO=AFCO,  and  the  A  COF 
==  A  FOB. 

Put  A  AFO=a;  then  A  FCO=a.  Also,  put  A  COF=b, 
as  represented  in  the  figure. 

Because  CB  is  bisected  in  F,  the  A  ACF  is  half  of  the  whole 
A  ABC.  Because  ^C  is  bisected  in  F,  the  A  BFC is  half  the 
whole  A  ABC. 

That  is,     2«+6==25+a. 

Whence,  a=b,  and  the  four  triangles  above  the  point 

0  are  equal  to  each  other. 

Let  the  area  of  the  A  AD  0  be  represented  by  x,  and  the  area 
oiDOBhjy. 

Now  taking  COD  as  the  base  of  the  triangles,  we  have 
2a     :    X     :    :     CO     :     OD 


Also, 


25= 
14 


2a 


CO 


OD 


210 


ROBINSON'S   SEQUEL. 


Whence, 
Therefore, 


AD=DB, 


2a     :     y.     Or,  .c=y. 
Q.  E.  I). 


(14.)  The  three  straigJd  lines  which  bisect  the  three  angles  of  a 
triangle,  meet  in  the  saine  point. 

Let  ABO  be  the  A, 
bisect  two  of  the  angles 
A  and  C — the  bisecting 
lines  will  meet  in  the 
same  point  0.  Join  OB; 
we  are  required  to  demon- 
strate that  OB  bisects  the 
angle  B. 

From  0,  let  fall  the  perpendiculars  on  to  the  sides.  The  two 
right  angled  A's  A  OH  and  A  00,  are  equal  in  all  respects,  be- 
cause they  have  the  same  hypotenuse  A  0,  and  equal  angles  by 
construction.  In  the  same  manner  we  jirove  that  the  A  CGO 
=  £\00L  Whence,  00=  OL  But  (7(9=  0//;  therefore, 
0H=  01. 

Now  in  the  two  right  angled  triangles  OHB  and  OIB,  we  have 
0H=  01,  and  OB  common,  therefore,  the  triangles  are  equal, 
'dridIIBO=OBL     Q.  E.  D. 

(15.)  The  two  triangles  formed  by  drawing  straight  lines  from 
any  point  within  a.  parallelogram  to  the  extremities  of  the  opposite 
sides,  are  together  half  the  parallelogram. 

Let  ABD  (7  be  a  parallelogram,  E  any 
point  within. 

We  are  to  show  that  the  triangles  AUB, 
CJED,  are  together  half  the  parallelogram. 

Through  the  point  £  draw  a  line  par- 
allel to  AB  or  CD,  thus  forming  two  parallelograms. 

The  A  AUB  is  half  the  lower  para,llelogram,  and  the  A  CUD 
is  half  the  upper  parallelogram  ;  therefore,  the  sum  of  the  two 
A's  is  half  the  whole  parallelogram.     Q.  E.  D. 

(16.)  The  figure  formed  by  joining  the  points  of  bisection  of  the 
aides  of  any  trapezium,  is  a  parallelogram. 


•    •      ^     .Ar 


GEOMETRY. 


211 


Let  AB  CD  ho  a  trapezium. 
Draw  the  diagonals  ^  (7,  JBD. 
Bisect  the  sides  in  a,  b,  c,  and 
(/.  Join  abed.  We  are  to  prove 
that  this  figure  is  a  parallelo- 


ABD  is  a  A  whose  sides  are 
bisected  in  a  and  b  ;  therefore, 
tlie  A  Aba  is  equiangular  to  the  A  ABD,  (th.  19,  b  ii),  and  ab 
is  parallel  to  BD,  and  by  (th.  18,  b.  ii),  ab=\BD.  In  the  same 
manner  we  can  prove  that  dc  is  parallel  to  BD  and  equal  to  half 
of  it.  Consequently  ab  and  dc  are  parallel  and  equal.  There- 
fore, by  (th.  23,  b.  i),  the  figure  abed  is  a  parallelogram.    Q.  E.D. 


(17.)  If  squares  be  described  on  the  three  sides  of  a  right  angled 
triangle,  and  the  extremities  of  the  adjacent  sides  be  joined,  the  triangles 
so  formed  are  equal  to  the  given  triangle,  and  to  each  other. 

LetJJ5(7be 

the  given  right 
angled  triangle 
and  construct 
the  figure  as 
here  represent- 
ed. It  is  ob- 
vious that  the 
vertical  right 
angled  A  ^l-^ff^ 
is  equal  to 
ABC. 

Draw  AV 
perpendicular 
to  BC,  and 
call  it  X.  We 
now  pro})ose  to 
show  that  HO 
=^x.  BD  is 
produced  to  G,  the  angles  VBOojidi  ABffare  right  angles,  and 


Il2  ROBINSON'S  SEQUEL. 

from    these   equals   take   away  the  common    part   ABG;    thus 
showino-  th^t  ABV=BBG. 

o 

The  two  right  angled  triangles  AB  V,  JIB  G  are  equal,  because 
they  have  equal  angles,  and  the  hypotenuse  AB=  the  hypote- 
nuse JIB,  because  they  are  sides  of  the  same  square.  Therefore, 
ffG=A  V,  and  if  one  is  in  value  x,  the  other  has  the  same  value. 

Now  we  designate  any  side  of  the  square  on  BC  by  a,  then 
twice  the  area  of  the  A  AB  C  is  ax,  and  the  double  area  of  the 
triangle  HBD  is  obviously  ax. 

Therefore,  HBD  is  equal  in  area  to  ABC. 

In  the  same  manner  we  can  prove  that  FCE:=ABC.     Q.E.D. 

(18.)  If  squares  he  described  on  the  hypotenuse  and  sides  of  a  right 

anffled  tnangle,  and  the  extremities  of  the  sides  of  the  former,  and  the 

adjacent  sides  of  the  others  he  joined,  the  sum  of  the  squares  of  Che  lines 

joining  them  ivill  he  equal  to  five  times  the  square  of  the  hypotenuse. 

(See  figure  to  the  last  Theorem.) 

In  the  right  angled  triangle  HGD,  we  have 

x'-+{BG+ay={HDY  (1) 

In  the  right  angled  triangle  PFE,  we  have 

x-+{PC^y={FEY  (2) 

Expanding  (1)  and  (2),  and  observing  that  GB-=BV,  PC= 
CV,  we  shall  have 

x^-\-{BVy-\-2a(BV)-{-a^=(IIDy         (3) 

And  x''-{-(FCy+2a{FC)+a-=:(FFy         (4) 

By  adding  (3)  and  (4),  and  observing  that  x- -\-(B  V)"  =b^ , 
andic2+(PC)2=c2,  then 

(J2^^2  )^2a(i?  r+  VC)-{-2a^=(IIFy+(FCy 
That  is,  a2_j_2rt(o)_j_2«2^ 

Or,  5a^  =  (IIDy+(FCy 

Scholium,  The  sum  of  the  squares  of  the  sides  of  the  last 
figure  is  8a  ^. 

(19.)  The  vei'ticol  angle  of  an  ohlique-angled  triangle,  inscribed  in  a 
circle,  is  greater  or  less  than  a  right  angle,  hy  the  angle  contained  be- 
tween the  base  and  the  diameter  draumfrom  the  extremity  of  tJie  base. 


* 


GEOMETRY. 


213 


Let  AE  C  be  a  A  liav- 
ing  the  angle  A  CB  grea- 
ter than  a  right  angle, 
and  describe  a  circle  a- 
bout  it.  From  one  ex- 
tremity of  the  base  as  B 
draw  the  diameter  BD. 

The  angle  DBC  is  a 
right  angle,  because  it  is 
in  a  semicircle.  The  ver- 
tical angle  A  CB  is  grea- 
ter than  a  right  angle 
hj  ACB;  but  ACD  is 
equal  ABD,  because 
each  is  measured  by  half  the  arc  AI).  Therefore  ACB  is  greater 
than  a  right  angle  by  ABD. 

Next  let  A'CB  be  the  A ;  the  angle  A'CB  is  less  than  a  righ^ 
angle  by  the  angle  DCA'=DBA\  hecause  each  is  measured  by 
half  the  arc  DA'.     Therefore,  the  vertical  angle,  (fee. 

(20.)  If  the  base  of  any  triangle  he  bisected  by  the  diameter  of  its 
circumscribing  circle,  and  from  the  extremity  of  that  diameter,  a  per- 
pendicidar  be  let  fall  upon  the  longer  side,  it  ivill  divide  that  side  into 
segmerds,  one  of  which  will  be  equal  half  the  sum,  and  the  other  half  the 
difference  of  the  sides. 
Let  .42? (7 be  the  A, 
bisect  its  base  by  the 
diameter  of  the  circle 
drawn  at  right  angles 


to  AB. 

From  the  center  0 
let  fall  Om  at  right 
angles  to  A  C,  it  will 
then  bisect  ylC  From 
the  extremity  of  the 
diameter  B,  draw  Bfh 
perpendicular  to  ^1 C, 
and  consequently  par- 
allel to  Om.  Produce 


214  ROBINSON'S  SEQUEL. 

Hh  to  M  and  join  ML.  Complete  and  letter  the  figure  as 
represented. 

The  two  triangles  Aah  and  Hha  are  equiangular.  The  angle 
a  is  common  to  them,  and  each  has  a  right  angle  by  construction, 
therefore  the  angle  H=^  the  angle  A.  But  equal  angles  at  the 
circumference  of  the  same  circle  subtend  equal  chords,  (th.  2,  b. 
iii ;}  therefore  CB^=ML.  The  angle  HML  is  a  right  angle,  be- 
cause it  is  in  a  semicircle,  therefore  ML  is  parallel  to  AC,  andJfZ 
is  bisected  in  n. 

Now  Am^\A C.     nL^md^ \ML=\B C. 

Therefore  by  addition,  Am-\-md=\(AC-\-CB,) 

Or,  Ad=\(AC-^CB.)        Q.  E.  D. 

Cor.  If  Ad  is  the  half  sum  of  the  sides,  dc  or  Ah  must  be  the 
half  difference  ;  for  the  half  sum  and  half  difference  make  the 
greater  of  any  two  quantities. 


(21.)  A  straight  line  dv&wn  frmi  the  vertex  of  an  equilateral  trian- 
gle, inscribed  in  a  circle,  to  any  point  in  the  opposite  circumference,  is 
equal  to  the  two  lines  together,  which  are  dratvn  from  the  extremities 
of  the  base  to  the  same  point. 

Let  ^i>6'be  the  e- 
quilateral  A  in  a  cir- 
cle. Take  I>  any  point 
in  the  arc  between  i> 
and  C,  and  join  A.D, 
BD,  andi)a 

Designate  each  side 
of  the  given  triangle 
by  a. 

Now  ABDC  is  a 
(juadrilateral  in  a  cir- 
cle, AD  is  one  diago- 
nal and  BC  iho,  otlier, 
and  by  (th.  21,  b.  iii) 
\ve  have 

u(AD)=a{BD)+a{DC,) 

Diyiding  by  a,  and     AD=:BD-^DC.         Q.  E.  D. 


GEOMETRY.  215 

(22.)  The  straight  line  bisecting  any  angle  of  a  triangle  inscribed 
in  a  given  circle,  cvis  the  circumference  in  a  "point  lohich  is  equidistant 
from  the  extremities  of  the  sides  opposite  to  the  bisected  angle,  and 
from  the  center  of  a  circle  inscribed  in  the  triangle. 

(See  the  figure  to  the  last  Theorem.) 

The  angle  BAD  is  measured  by  half  the  arc  BD,  (th.  8,  b.iii) 
and  the  angle  DA  C  is  measured  by  half  the  arc  D  C ;  therefore, 
if  BAD=DAC,  the  arc  BD  must  equal  the  arc  DC. 

(23.)  If  from  the  cerder  of  a  circle  a  line  be  drawn  to  any  point  in 
the  chord  of  an  arc,  the  square  of  that  line,  together  with  the  rectangle 
contained  by  the  segments  of  the  chord,  will  be  equal  to  the  square 
described  on  the  radius. 

(See  the  figure  to  the  21st  Theorem.) 

From  the  center  0  draw  0  F  to  any  point  in  A  0,  and  through 
the  point  Fdraw  nm  at  right  angles  to  OV,  and  join  Om ;  then 
0  Vm  is  a  right  angled  triangle.  Therefore,  (  0  F)^-|-(  Vmy  = 
{Om)\  But  (Vmy  =  (nV)  (Vm)=(AV)  (VC),  (th.  17,  b. 
iii.)     Therefore,  by  substitution, 

{  OVy-\-{AV)  (VC)=(  Omy .     Q.  E.  D. 

(24.)  If  two  poiMs  be  taken  in  the  diameter  of  a  circle,  equidistant 
frmn  the  center,  the  sum  of  the  squares  of  the  two  lines  drawn  frmn 
these  points  to  any  point  in  the  circumference  will  be  always  the  same. 

Let  C  be  the  cen- 
ter of  a  circle,  and  A 
any  point  in  the  cir- 
cumference. CA= 
r,  the  radius. 

Put  AD=y,  DO 
=ar,  and  CB  and  CG 
each  =a.  Then  BD 
=(x — a),  and  DG 
=(x+u). 

Now  in  the  triangle 
ADB  we  have  y^-{-(x^ay=(ABy. 

And  in  the  triangle  ADG,     y^-^(x-^ay—(AGy 


216 


ROBINSON'S   SEQUEL. 


By  expanding  and  adding,  we  find 

The  triangle -4i> (7  gives    2?/^ -\-2x^ —^r'^ ;  therefore, 
2r^+2a^=(ABy-\-(AGy . 

Because  the  first  member  of  this  equation  is  the  same  for  all 
values  of  x  and  y — that  is,  because  it  is  invariable  ;  therefore  the 
second  member  must  also  be  invariable.      Q.  E.  D. 


(25.)  If  on  the  diameter  of  a  semicircle  two  equal  circles  be  described^ 
and  in  ike  space  included  by  the  three  circumferences,  a  circle  be  in- 
scribed, its  diameter  will  be  two-thirds  the  diameter  of  either  of  the 
equal  circles. 

It  is  sufficient  to  represent  a  portion  of  the  figure. 

Let  B  be  the  cen- 
ter of  the  semicircle, 
and  BA  the  diame- 
ter of  one  of  the  e- 
qual  circles,  and  E 
the  center  of  the  cir- 
cle sought — BD  be- 
ing at  right  angles  to 
AB  from  the  point  B. 

Put  CB=r,  and 
DE=x.  Then  BD 
=2r,  BE=:^2r-—x, 
and  CE=r-\-x. 

Now  in  the  right 
angled  triangle  BEC, 
we  have 

That  is. 

By  expanding. 

Reducing, 

Whence, 


{CBY+{BEY=(CEY. 

r2J^[2r—xY={r+xy. 

7.2 +4r2  ^Arx+x""  =r^  J^^rx+x^ . 

x=§r.         Q.  E.  D. 


(26.)  If  a  perpendicular  be  drawn  from  the  vertical  angle  of  any 
triangle  to  the  base,  the  difference  of  the  squares  of  the  sides  is  equal 
to  the  differeyice  of  the  squares  of  the  seginerUs  of  the  base. 


GEOMETRY.  217 

Let  ABC  be  any  triangle.  Let  fall  AD 
perpendicular  to  the  base.  Now  the  two 
right  angled  triangles  give  us 

(ADy-{-{BDy={ABy. 

And     (ADy-\-(I)C)^={AC)K 

By  subtraction,  (BDy—{D Cy^jABy'—jA C) ^ .     Q.  E.  D. 

By  factoring,  {BD-{-DC){BD—DC)={AB-\-AC){AB—AC.) 
By  observing  that  (BD-\-J)C)=BC,  and  converting  this  equa- 
tion into  a  proportion,  we  have 

BC    :     (AB+AC)     :    :     (AB—AC)     :     (BD—DQ.) 
(This    is    Prop.  6,  Plane  Trigonometry,  page  149,  Robinson's  Geometry.) 

ScHo.  This  proportion  is  true  whatever  be  the  relation  of  AB 
to  AC.    It  is  true  then  when  AB=AC.    Making  this  supposition, 
then  BD  becomes  equal  to  D  C,  and  the  proportion  becomes 
BC     :     AB+AC    :    :     0     :     0. 

Now  (AB-^AC)  being  two  sides  of  a  triangle  are  greater  than 
the  third  side  BC ;  therefore  the  last  zero  is  greater  than  the  first,  an 
apparent  absurdity. 

But  this  is  no  more  than  saying  that  zero  divided  by  zero  can 
have  a  positive  quotient — for  we  can  subtract  zero  from  zero  as 
many  times  as  we  please,  and  still  have  zero  left. 

The  proportion  is  obviously  true,  for  0  times  BC  is  equal  to  0 
times  {^AB-\-AC.)  Indeed  0  may  be  to  0,  as  a  to  any  quantity 
Avhatever. 

(  27.)  The  square  described  on  the  side  of  an  equilateral  triangle  is 
equal  to  three  times  the  square  of  the  radius  of  the  circumscribing 
circle. 

Let  ABC  be  the  equilateral  tri- 
angle. Let  fall  the  perpendicular 
AE  on  the  base  ;  it  will  bisect  the 
base.  Draw  BD  bisecting  the  an- 
gle at  B.  D  will  be  the  center  of 
the  circumscribing  circle,  and  AD  or 
BD  will  be  the  radius. 

We  are  to  prove  AD=BD^  and 
find  the  value  of  BD  in  terms  of  AB. 


218  KOBINSON'S  SEQUEL. 

Each  angle  of  an  equilateral  triangle  is  60^,  (^  of  180°.) 
If  we  bisect  these,  each  division  will  be  30°. 
Hence  BAD=^30°,  and  ABJ)=30° ;  therefore,  AD=BJ). 
Put  AB=2a,  then  BE=a.     Also  put  BD==x,  then  DE=lx* 
Now  in  the  riifht  an^^led  trianojle  BDE,  we  have 

Whence,  ^a^^-dx"".     But   ^w"  ={ABY . 

Therefore,  {ABY^7>{BDY,     Q.  E.  D. 

(28.)  The  sum  of  the  sides  of  an  isosceles  triangle,  is  less  than  the 
sum  of  any  other  triangle  on  the  same  base,  and  between  the  same 
parallels. 

Let  ABC  be  the  isosceles  tri- 
angle. AB=AC.  Throuorh  the 
point  A  draw  GAH  parallel  to 
BC. 

Take  G  any  other  point  on  the 
line  GH,  and'draw  i?6^and  GO. 

We  are  to  show  that  AB-\-AO 
are  less  than^C 6^-|-  G C.  Produce 
AB  to  D,  making  AJ)=AB,  or 
AC. 

Then  by  reason  of  the  parallels  GH  and  B  C,  the  angle  BAH 
is  equal  to  the  angle  ABO,  and  IIAC=ABC. 

Because  AD=AC,  the  anole  ADII=  the  anerle  ACH. 

Whence  the  two  triangles  AD  If  and  ACH,  are  equal  in  all 
respects,  and  GB  is  perpendicular  to -DC;  whence  any  point  in 
the  line  GHis  equally  distant  from  the  two  points  D  and  C. 

Now  the  straight  hne  BI)=BA-\-AC,  and  because  I>G=GC, 
B  G-\-  GB=  GB+  GC.  But  X>  6^+  GB,  the  two  sides  of  a  A  are 
greater  than  the  third  side  jDB  ;  therefore,  GB-\-GC  are  greater 
than  BD,  that  is,  greater  than  B^A-fAC.     Q.  E.  D. ' 

*This  mjglit  not  be  admitted,  at  tlie  same  time  the  reader  would  readily 
admit  that  BE  was  one-half  AB.  ABE  is  aright  angled  triangle,  one  angle 
being  30  deg.  the  side  opposite  that  angle  is  half  the  hypotenuse,  and  this  is 
a  general  truth.  Now  the  angle  DBE  equals  30  deg.,  therefore  DE  is  half 
BD. 


GEOMETRY. 


219 


GEOMETRICAL  CONSTRUCTIONS. 


(29.)  In  any  triangle,  given  one  angle,  a  side  adjacent  to  the  given 
angle,  and  the  difference  of  the  other  two  sides,  to  construct  the  triangle. 

Let  ^^  represent  the  giv- 
en side,  and  from  one  ex- 
tremity as  Ay  make  the  an- 
gle BAC=  to  the  given 
angle,  (prob.  5,  b.  iv.) 

Take  AJ)=  to  the  given 
difference  of  the  sides,  and 
join  DB.  From  the  point  B  make  the  angle  DBO  equal  to  the 
angle  BDC,  then  CB=CD,  smd  AD  is  the  given  diflPerence  of  the 
sides,  and  ABC  is  the  triangle  required. 


(30.)  In  any  triangle,  given  the  base,  the  sum  of  the  other  two  sides, 
and  the  angle  opposite  the  base,  to  construct  the  triangle. 

Draw  AC  equal  to 

the  sum  of  the  sides. 

From  the  point  ^  as  a 

center,  with  a  radius  e- 

qual  to  the  given  base 

AB,  describe  an  arc  as 

represented  in  the  fig- 
ure. 

From  the  point  C  in 

the  line  A  C,  make  the 

angle   ACB    equal  to 

half  the  given  angle. 
If    the    problem    is 

possible,  this  line    CB 

will  cut  the  circular  arc 
in  two  points,  B  and  B'.  From  B  and  B'  make  the  angles  CBD 
and  CB'D',  each  equal  to  the  angle  at  C.  Join  AB,  AB',  and 
either  A  ABD  or  AB'D',  fulfils  the  required  conditions. 

For   CD=DB,  and    CD'=B'D',  (because  they  are  sides  of  a 
A  opposite  equal  angles,)  therefore  AD-{-DB=mA  C ;  also  AD'-^- 


220  ROBINSON'S  SEQUEL. 

D  B'-^AC.     The  angle  ADB  is  double  the  angle  C,  (th.  11,  b. 
i,)  therefore  it  is  the  angle  required. 

'  (31.)  In  any  triangle,  given  the  base,  the  angle  opposite  to  the  base, 
and  the  difference  of  the  other  two  sides,  to  construct  the  triangle. 

Subtract  the  given  angle  from  180°,  and  divide  the  remainder 
by  2,  designating  the  result  by  a. 

Draw  an  indefinite  line  as  AC,  (see  figure  to  29,)  and  take 
AD  equal  to  the  given  difference  of  the  sides. 

From  the  point  D,  make  the  angle  CDB-=-a. 

From  ji  as  a  center,  with  a  radius  equal  to  the  given  base  AB, 
strike  an  arc,  cutting  DB  in  B. 

At  J5make  the  angle  DBC=a;  then  DC==BC,  and  ABC 
will  be  the  triangle  required. 

(^^2.)  In  any  triangle,  given  the  base,  the  perpendicular,  and  the 
angle  opposite  to  the  base,  to  construct  the  triangle. 

Draw  AB  equal  to  the  given 
base,  and  D  C  parallel  to  it  at 
the  given  perpendicular  dis- 
tance. 

On  the  other  side  of  the  base 
AB,  make  the  angle  BAG  e- 
qual  to  j^art  of  the  gi^i^en  angle, 
and  ABG  equal  to  the  reinain- 
ing  part,  thus  forming  the  A 
AGB.  About  the  A  ABG, 
describe  a  circle  cutting  DCm 
the  points  D  and  C.  Join  A  C, 
CB,  and  xiCB  is  the  triangle  required. 

The  angle  BCG=BAG,  (th..9,  b.  iii,  scho.),  and  the  angle 
ACG=ABG.  Therefore  by  addition,  ACB=BAG+ABG; 
that  is,  ACB-=  the  given  angle. 

The  triangle  ADB  will  also  answer  the  conditions  ;  for  ACB 
=ADB. 

(33.)  In  any  triaiigle,  given  the  base,  the  ratio  of  the  two  sides, 
a7id  the  line  bisecting  the  vertical  angle,  to  construct  the  triangle. 


GEOMETRY. 


221 


Draw  the  base  EG,  and  bisect  it  in  i). 
Draw  DB  at  right  angles  to  EG. 

Divide  EO  in  the  point  /,  so  that  ^/ shall 
be  to  IG  in  the  ratio  of  EH  to  HG, 

Find  IB  of  such  a  ralue  that 

HI    :     GI    :    :     EI    :     IB. 

The  three  first  terms  are  given  ;  therefore 
the  fourth  is  known.  From  /  as  a  center,  with  the  distance  IB 
as  radius,  strike  an  arc,  c^utting  DB  in  B.  Join  BI  and  produce 
it  to  H,  making  ZST  equal  to  the  given  distance.  Join  EH,  HG^ 
and  EHG  is  the  A  required. 

Because  HIy^IB=EIy^IG,  a  circle  which  passes  through  the 
points  E,  B,  and  G,  will  also  pass  through  the  point  H,  and  the 
angle  EHI=IHG,  and  for  that  reason  EH  \  HG  \  \  EI  \  IG, 
as  required.     (See  th.  25,  b.  ii.) 


(34.)  To  draw  a  straight  line  through  any  given  j^oint  within  a 
triangle  to  meet  the  sides ,  or  the  sides  produced,  so  that  the  given  point 
shall  bisect  the  line  so  drawn. 

Let  ABO  be  the  A,  and 
/^  the  given  point  within  it. 

Through  F  it  is  required 
to  dra2v  the  straight  line  gl,  so 
thai  gP  shall  be  equal  PL 

From  P  draw  PH  paral- 
lel to  AB.     Take  gH=AH. 

Join  gP  and  produce  it  to  I,  and  gl  is  the  line  required. 
P^is  parallel  to  Al, 

gH    :     HA     :    :    gP     :     PL  ^ 

But    gH=HA  ;  therefore  gP=Pl. 

ScHO.  Had  we  taken  Hg  double  of  AH,  then  gP  would  have 
been  double  of  PI,  and  we  might  have  required  gP  to  be  any 
number  of  times  PL 


Because 


(35.)  Find  the  square  roof  of  \3  or  any  other  number,  by  a  geO' 
metrical  construction. 


222  ROBINSON'S  SEQUEL. 

Divide  the  number  into  any  two  factors,  (say  2  and  6},)  add 
the  factors  to<rether,  for  the  diameter  of  a  circle. 

Take  the  half  sum  of  the  two  factors  for 
the  radius  of  a  circle,  and  describe  the  cir- 
cle as  represented  in  the  margin. 

Let  AB  be  a  diameter,  and  take  AD  for 
one  factor,  and  BB  for  the  other  ;  and 
through  I),  draw  FJS  at  right  angles  to^^. 
JDU  or  DF represents  the  square  root  required. 

In  the  present  example,  if  ^i>=2  and  DB=Q^;  then  the 
length  of  DE  applied  to  the  same  scale  will  show  the  square  root 
of  13.     Because  ABxDB={DJSy. 

When  the  two  factors  are  very  nearly  equal,  D  will  be  very 
near  the  center  of  the  circle,  and  DF  will  be  very  nearly  the  ra- 
dius of  the  circle, — always  a  little  less,  unless  the  factors  ar«^ 
absolutely  equal ;  in  that  case  each  one  is  a  root.  On  this  prin- 
ciple toe  extracted  square  root  in  the  first  part  of  this  volume. 

Observe  the  A  F>  CE.  CE  is  the  half  sum  of  the  two  factors, 
and  DC  is  their  half  difference. 

Also,  DE  is  the  sine  of  the  arc  AE^  and  DC  is  the  cosine  of 
the  same  arc  ;  therefore,  ive  can  if  we  desire  it,  bring  in  the  aid  of 
a.  table  of  natural  sines  and  cosities. 

But  the  tables  of  natural  sines  are  adapted  to  radius  unity ;  lieie 
the  radius  is  4|,  therefore  to  have  corresponding  values  of  CD 
and  DEy  we  have  this  proportion, 

^     :     \     :    :     ^     :     .52941, 

The  result  of  this  proportion  carried  to  the  table  of  natural  sine ; 
gives  .848365  for  the  corresponding  cosine,  and  this  multiplied  by 
4J,  gives  3.605551  for  the  square  root  of  13. 

Another  Construction. 

(36.)  Let  it  be  required  to  find  the  square  root  of  250,  (or  any 
other  number,)  by  a  geometrical  construction. 


# 


GEOMETRY. 


223 


Divide  the  number  into  two  factors. 
Let  one  factor  be  represented  by  AB, 
the  other  hy  AC;  BC  being  their 
diflference.  On  the  difference  as  a 
diameter,  describe  a  circle. 

From  the  extremity  A,  draw  AD 
touching  the  circle.  AD  represents 
the  square  root  required. 

By  (th.  18,  b.  iii,  scho.  1), 
ABXAC={AD)K 

Therefore  AD  is  the  square  root 
of  the  product  of  the  two  factors 
AC  2iTidiAB. 

Reinark.  When  the  two  factors  are  nearly  equal,  the  circle  will 
be  very  small,  and  AD  will  be  very  nearly  Ao.  But  xio  is  the 
half  sum  of  the  factors  AB  and  A  C,  hence  we  know  that  the 
square  root  of  the  product  of  two  factors  is  always  a  little  less  than 
their  half  sum,  unless  the  factors  are  absolutely  equal. 

In  the  proposed  example,  we  divide  250  into  the  two  factors, 
25  and  10 — their  diflference  is  15.  Hence  7^  is  the  radius  of  the 
circle.  Take  7|-  from  any  scale  of  equal  parts  in  the  dividers, 
and  describe  a  circle. 

Draw  any  diameter  as  B  C,  and  produce  it  to  A,  making  AB=^ 
10.  From  A  draw^i)  to  touch  the  circle  ;  take  that  distance  in 
the  dividers  and  apply  it  to  the  scale,  and  the  result  will  be  the 
square  root  of  250. 

The  practical  difficulty  in  this  construction  is  to  decide  exactly 
where  the  point  D  is,  therefore  the  first  method  of  construction 
is  the  best. 

Geometrical  constructions  are  not  to  be  relied  upon  for  numer- 
ical accuracy,  but  they  are  invaluable  to  impress  theory,  and  are 
sure  guides  to  numerical  operations. 

Scho.  If  it  were  required  to  make  a  square  equal  to  a  given 
rectangle,  either  of  the  two  preceding  constructions  may  be  applied. 
Let  ^C  be  one  side  of  the  rectangle,  AB  the  other;  then  AD 
will  be  a  side  of  the  required  square. 


224  ROBINSON'S  SEQUEL. 

PROBLEMS. 

The  following  problems  do  not  admit  of  geometrical  construc- 
tions, in  the  sense  of  some  of  the  preceding — they  require  alge- 
bra applied  to  geometry. 

We  take  the  problems  from  Robinson's  Geometry,  pages  105 
to  109.     For  theory,  the  reader  must  look  elsewhere. 

We  omit  the  first  two  problems,  and  number  them  as  they  are 
numbered  in  the  geometry. 

PKOBLEM  3. 

In  a  triangle  J  having  given  the  sides  about  the  vertical  angle,  and 
the  line  bisecting  that  angle  and  terminating  in  the  base,  to  find  the 
base. 

Let  ABC  he  the  A,  and  let  a  circle  be 
circumscribed  about  it.  Divide  the  arc  AliJB 
into  two  equal  parts  at  the  point  jE,  and  join 
£JC.  This  line  bisects  the  vertical  angle, 
(th.  9,  b.  iii,  scho.)     Join  BU. 

Put  AD=x,' JDB=g,  AC=a,  CB=b, 
CD=c,  and  DjE=w.  The  two  A's,  ADC 
and  JiJB  C,  are  equiangular  ;  from  which  we  have, 

w-\-c     :     b     :    :     a     :     c.     Or,  cw-^c^  =ab.     (1) 

But  as  JtJC  and  AB  are  two  chords  that  intersect  each  other  in 
a  circle,  we  have,  cw=xy         (th.  17,  b.  iii.) 

Therefore,  xg-\-c'^=ab  (2) 

But  as  CD  bisects  the  vertical  angle,  we  have, 

a     :    b     :    :    X     :    y        (th.  23,  b.  ii.) 

Or,  x=^-l  (3) 

0 


Hence,         -y' -\^^ zz=ab :  or,  y=J( 


b"     '  '  "      \j  ^ 


And,  x=?J6»-el* 

b^  a 

Now  as  X  and  y  are  determined,  the  base  is  determined. 
N.  B.  Observe  that  equation  (2)  is  theorem  20,  book  Hi. 


GEOMETRY.  225 

PROBLEM  4. 

To  determine  a  triangle,  from  the  base,  the  line  bisecting  the  ver- 
tical dngle,  and  the  diameter  of  the  circumscribing  circle. 

Describe  the  circle  on  the  given  diameter 
AB,  and  divide  it  in  two  parts,  in  the  point 
i>>  so  that  ADxI>B  shall  be  equal  to  the 
square  of  one-half  the  given  base. 

Through  D  draw  ED  G  at  right  angles  to 
AB,  and  EG  will  be  the  given  base  of  the  A- 

Put    AD=^n,  DB:=m,  AB=d,  DG=b. 

Then,  n-\-m=d,  and  nm=:b^  ;  and  these  two  equations  will 
determine  n  and  m  ;  and  therefore,  n  and  m  we  shall  consider  ijis 
known. 

Now  suppose  EHG  to  be  the  required  A,  and  join    TUB  and 

HA.     The  two  A's  AHB,  DBF,  are  equiani^ailar,  and  therefore, 

we  have, 

AB     :     HB     :    :     IB     :     DB. 

But  BI  is  a  given  line,  that  we  will  represent  by  c  ;  and  if  we 
put  IB=iv,  we  8hall  have  IIB=c-\-w ;  then  the  above  proportion 
becomes,  d     :     c-f-w     :    :     w     :     m. 

Now  w  can  be  determined  by  a  quadratic  equation  ;  and  there- 
fore IB  is  a  known  line. 

In  the  right  angled  A  DBI,  the  hypotenuse  IB,  and  the  base 
DB,  are  known  ;  therefore,  I>I  is  known,  (th.  36,  h.  i);  and  if 
i)/is  known.  Eland  IG  are  known. 

Lastly,  let  EH—x,  IIG=y,  and  put  EI=p,  and  IG=q. 

Then  by  theorem  20,  book  iii,    pq-\-c^:=xy  (1) 

But,  X     :    y     :    :    p     :     q  (th.  25,  b.  ii.)     ^ 

Or,  x^^l  (2) 

q 

And  from  equations  (1)  and  (2)  we  can  determine  x  and  y,  the 
sides  of  the  A  ;  and  thus  the  determination  has  been  attained, 
carefully  and  easily,  step  by  step. 

PROBLEM  5. 
Three  equal  circles  touch  each  other  externally,  and  thus  inclose  one 
acre  of  ground  ;  what  is  the  diameter  in  rods  of  each  of  these  circles  ? 
15 


226 


ROBINSON'S  SEQUEL. 


Draw  tliree  equal  circles  to  touch  each 
other  externally,  and  join  the  three  centers, 
thus  forming  a  triangle.  The  lines  joining 
the  centers  will  pass  through  the  points  of 
contact,  (th.  7,  b.  iii.) 

Let  H    represent    the   radius    of  these 
equal  circles  ;  then  it  is  obvious   that  each 
side  of  this  A  is  equal  to  2JR.     The  triangle  is  therefore  equilat- 
eral, and  it  incloses  the  given  area,  and  three  equal  sectors. 

As  each  sector  is  a  third  of  two  right  angle*,  the  three  sectors 
are,  together,  equal  to  a  semicircle  ;  but  the  area  of  a  semicircle, 

whose  radius  is  M,  is  expressed  by  -  -*^--  (th.  3,  b,  v,  and  th.   1, 

b.  v);  and  the   area  of  the  whole  triangle  must  be -f"^^^  * 

but  the  area  of  the  A  is  also  equal  to  Ji  multiplied  by  the  per- 
pendicular altitude,  which  is  BJS. . 


Therefore, 
Or, 

Hence, 


i22j3=---+160. 

2 

723(2^3— rt)=320. 
320 


320 


2^3—3.1415926     0.3225 
i?= 31.48-1- rods  for  the  result. 


:992.248. 


PROBLEM  6. 

In  a  right  angled  triangle,  having  given  the  hose  and  the  sum  of  the 
perpendicular  and  hypotenuse,  to  find  these  two  sides. 

Let  ABC  he  the  A.  Put  CB= 
h,  AB+AC=a,  AB=x  ;  then  AC 
=a — X. 

By  (th.  36,  b.  i), 

a'—b^ 


Whence, 


2a 


Now  the  numerical  value  of  x  being  known,  the  triangle  can 
be  constructed  geometrically. 


GEOMETRY. 


227 


PROBLEM  7. 

Given  the  base  and  altitude  of  a  triangle^  to  divide  it  into  three 
equal  parts,  by  lines  parallel  to  the  base. 

Let  ^^  6"  represent  the  A.  Conceive 
a  perpendicular  let  drop  from  C  to  the 
base  AB,  and  represent  it  by  b.  Put 
2a=AJB.  Then  ab=  the  area  of  the 
triangle. 

Let  X  be  the  distance  from  C  to  FD ; 
then  by  (th.  22,  b.  ii),  we  have, 

x^      :     b^      :    :     ^ab     :     ab 
Whence,      x     ;     b     :    i     \     \     J^, 

If  X  represents  the  distance  from  C  to  GE,  then 

x^      :     b^     :    :     f«6     :     ab. 
Or,  X     :       b     :    :     Ji     t     J3,     ^=^ 

We  perceive  by  this  tliat  the  divisions  of  the  perpendicular  are 
independent  of  the  base,  and  that  we  may  divide  the  triangle  into 
any  required  number  of  parts,  m,  n,  p,  <fec.,  equal  or  unequal. 


PROBLEM  8. 

In  any  equilateral  triangle,  given  the  length  of  the  three  perpendic- 
ulars drawn  from  any  point  within,  ta  the  three  sides,  to  determiiie 
the  sides. 

Let  ABC  be  the  A-  We  have 
shown  in  this  volume  tliat  CD= 
PG+PH-{-PO=a. 

Put  AD  or  VB^x  ;  then  BC^ 
2x. 

Then  by  th-e  right  angled  A 
CDB,  we   have     a~-|-.r2=4x'",  or 

V3 


298  ROBINSON'S   SEQUEL. 

PROBLEM  9. 

In  a  right  angled  triangle^  having  given  the  base  (3),  and  the  dif- 
ference hettoeen  the  hypotenuse  and  perpendicular  (1),  to  find  these 
sides. 

(See  figure  to  Problem  6.) 

Let  (7j5=3.     AO—AB^l.     AB=x.     Then  ^(7=a:+l,  and 
ara+9=aj2+2a'+L     x=4. 

PROBLEM   10. 

In  a  right  angled  triangle,  having  given  the  hypotenuse  (5),  and 
thediference  between  the  base  and  perpendicular  (1),  to  determine  both 
of  these  tivo  sides. 

(See  figure  to  Problem  6.) 

Let      CB=x.    AB=x-\-l.     Then 

Or,     2x^+2x=24.     Whence,  x=3.     AB=4. 

PROBLEM  11. 

Having  given  the  area  or  measure  of  the  space  of  a  rectangle  in- 
scribed in  a  given  triangle,  to  determine  the  sides  of  the  rectangle. 

When  we  say  that  a  triangle  is 
given,  we  mean  that  the  base  and 
perpendicular  are  given. 

Let  ABC  be  the  triangle,  AB=h, 
CD=p,  CI=x  ;  then  ID=p—x. 

By  proportional  triangles  we  have 
GI    :     EF    :    :     CD     :    AB 

That  is,        X     :    EF    I    :    p     :    b.     EF=—. 

P 

By  the  problem  —(p—x\—a.     The  symbol  a  being  the  giv 
en  area. 

Whence,  x^—pxz=--^,     x=^p:l  ^\p^^?^. 


GEOMETRY.  229 


PROBLEM  12. 


In  a  triangle  having  given  the  ratio  of  the  two  sides,  together  with 
both  the  segments  of  the  base,  made  by  a  perpendictdar  from  the  vertical 
angle,  to  determine  the  sides  of  the  triangle. 

Let  ACB  be  the  A,  (see  last  figure.)  AD=a,  BDr=b,  and 
CD=x.     Then  AC=Ja^-{-x^,  and  CB=Jb^+x^. 

The  ratio  oi  AC  to  CB  is  given,  and  let  that  ratio  be  as  1  to 
r  ;  then 

Ja^+x^     :     j¥+x^     :    :     1      :     r. 

Whence,       «2_|_^2     .        52_j_^3     :    :     i     :     rK 

Or,  b^+x^=a^r''+r''x'' 

Or, 


a^r^-^b^ 


But  AC=:Ja^-\-x'^,  and  as  x^  is  now  known,  ^Cis  known. 

PROBLEM  13. 
In  any  triangle  having  given  the  base,  the  sum  of  the  other  two  sides 
and  the  length  of  a  line  drawn  from  the  vertical  angle  to  the  middle  of 
the  base,  to  find  the  sides  of  the  triangle. 

Let  ADE  be  the  A-     Suppose   C 
to  be  the  middle  of  the  base. 

Put  AC=a,  DO  or  CE=^b,  AE 
=x,  DA^AE=c  ;  then  DA:=c—x. 

Now  by  (th.  39,  b.  i),  we  have 
(DAy+{AEY=^(ACy-\-2{DC)~ 

That  is,  c2— 2car+2a;2=2a2_|-262. 

Or,  4a;2—4ca:+c2=:4a2 +462—^2. 

Zx—c=  ^4a2+4p— c2 . 

Whence  x  becomes  known,  and  consequently  the  sides  become 
known. 

PROBLEM  14. 
To  determine  a  right  angled  triangle,  having  given  the  length  of 
two  lines  dravm  from  the  acute  angles  to  the  middle  of  the  opposite 
sides. 


idO 


ROBINSON'S  SEQUEL. 


Let  ABC  be  the  triangle.     Letter 
it  as  represented — CE^=a,  AI>==b,  &c. 

Then      \^^'+y'=^''\ 

By  add.    5x^ +^'' =a^  ■\-b^  =din. 


3^ 


2 «2 


a* — m. 


x=Ja^ — m.     y==.Jb^ — m. 
~T~  "~3~" 

PROBLEM  15. 

To  determine  a  right  angled  triangle,  having  given   the  perimeter 
and  the  radius  of  its  inscribed  circle. 

Let  ^^C  be  the 
A,  OjE'the  radius 
of  the  circle. 

It  is  obvious  that 
AE=A£>,  CF= 
CD.  Put  AE=x, 
CF^y,  FB^T, 
^p=.  the  perime- 
ter. Then  by  the 
conditions, 
:t+y-fr=p       (1) 

From  the  right 
angled  A  ABC, 
we  have 

(x+rY+(y+TY=^(x+yy 
By  reduction, 

rx-\-ry-\-r^  =xy 

That  is,  (x-^y-\'r)r^=^rp^=xy 

Equation  (4)  expresses  the  area  of  the  triangle. 
From  (1),         x^'-^-^xy+y' 
From  (4),  Axy         = 


(2) 
(4) 


'2  — =o2 — 2»r-4-r^ . 


Apr 
By  subtraction,  x^  —2xy-\-y^  =p^  — 6pr-|-r» . 


GEOMETRY.  fftt 


Whence,  x — y=  ±  Jp  ^  — 6jt?r-[-r  - . 

But  x-{-y—p — r. 


Therefore,        x=z^{p—r)±^Jp''—Qpr+r^. 

PROBLEM  16 

To  determine  a  triangle ,  having  given  the  base,  the  perpendicular, 
and  the  ratio  of  the  two  sides. 

Let  ABO  be  the   A-      AB=b, 
CD=a,  J)B=x.     Then 


CB:=/a:^^+^. 
Let  the  given  ratio  of  the  sides  be 
as  fn  to  n  ;  then 


J(b — x)^-\-a^     :     ^a^-\-x'^     :     :        m     :     n. 
This  proportion  will  give   the  value  of  or,  then  AC  and  CB 
will  be  known. 

PROBLEM  17. 

To  determine  a,  right  angled  triangle,  having  given  the  hypotenuse, 
and  the  side  of  the  inscribed  square. 

Let  J[i) (7  be  the  A.     (See  last  figure.)     Put    CI—x,IE=ia, 
A  G=y,  and  A  C=h.      Then  by  proportional  triangles,  we  have 
CI    :     lU    :    :     JfJG     :     GA. 
That  is,  X     :      a       :    :       a       '.      y.  Whence,  2-5^=0^. 

In  the  right  angled  A's  AGE,  ECL  we  have 

AE^  Jf~+a^.     EC=  Jx^^a- . 
Observing   that  AE-\-EC=AC—b,  and  a~—xy,  we  perceive 
that 


Jx^-\-xy-\'Jy~-{-xy^h. 

Whence,  »Jx+Jy=.    A^^ . 

■_        7  3 

By  squaring,  x-\-y-\-^Jxy—-,!—- . 

x+y       ___ 

Put  (a?+y)=-5,  and  observe  that  ^Jxy=^Za  ;  then 
52 -f.  2^5=^2  _ 


232 


ROBINSON'S  SEQUEL. 


Whence,  «+a=  zt^a^-j-i^. 

Now  having  the  value  of  (x-^-y),  and  (ary)  the  separate  values 
of  X  and  y  can  be  determined,  which  is  a  solution  of  the  problem. 

PROBLEM  18. 

To  determine  the  radii  of  three  equal  circles,  inscribed  in  a  given, 
circle  to  touch  each  other,  and  also  to  touch  the  circumference  of  the 
given  circle. 

Let  AD 
B  be  the 
given  cir- 
cle. Di- 
vide the 
circumfe- 
rence 360 
deg.  into 
3  equal 
parts.  BD 
is  one  of 
those  parts 

120°  ;  then  the  arc  ^i)=60°.      A   circle   inscribed  in  the    A 
COE,  will  be  one  of  the  equal  circles  required. 

Let  A  0=a,  AH=x,  H  being  the  center  of  the  circle.  From 
H,  draw  i/F" perpendicular  to  CO,  then  AH=:HV. 

Hence  FIVz!=x,  OH=a — i?-,  and  0  F=  J- 0^,  because  the  an- 
gle F^0=30°.     (See  prop.  1,  plane  trig.,  page  139.) 

Now  by  the  right  angled  A  0  VH,  we  have 
{OV)^-[-{VHY={OHy. 

That  is,  (^^i::fy+a:2=(a— ar)«.  ^ 

Whence,  a;={273— 3)a. 

PROBLEM  19. 

In  a  right  angled  triangle,  hamng  given  the  periimter,  or  sum  of 
all  the  sides,  and  the  perpendicular  let  fall  from  the  right  angle  on  the 
hypotenuse,  to  determine  the  triangle,  that  is,  its  sides. 


GEOMETRY. 


238 


Let  ABC  he  the  A,  and  represent  its 
perimeter  byjo.  Put  AI>=:a,  AB:=x, 
A  C=y.     Then  B  C=zp—x—y, 

Because  BA  C  is  a  right  angle, 

x^-\-y^-=p^—2p{x-\-y)-\-x^-\.2xy+y 
And,  a{p — x — y)=xy 

Reducing  (1),  ^p{^-\-y)='P^-\-^y 

Double  (2),  'Hap — 2a{x-^^y)  =  '2.xy 

By  subtraction,   (2a-j-2p)  (^+y) — ^op=p^ 


Whence, 


x-\-y 


_p^-\-2ap 


(1) 

(2) 
(3) 
(4) 
(6) 

(6) 


Because  BC=p-x^,  BC=p-t±^l=  ^t.._ 

2«+%)      2rt+2/) 


OjB" 


From  (2)  we  observe  that  xy 

^    '  2a4-2p 

Equations  (6)  and  (7),  will  readily  give  x  and  y. 


(7) 


PROBLEM  20. 

To  detennine  a  right  angled  triangle,  having  given  the  hypotenuse 
and  the  difference  of  two  lines,  drawn  from  the  two  acute  angles  to 
the  center  of  the  inscribed  circle. 

Let  ABC  be  the 
A,  0  the  center  of 
the  inscribed  circle; 
then  A  0  bisects  the 
angle  CAB,  and 
CO  bisects  the  an- 
gle  C. 

The  angle  A  OH, 
being  the  exterior 
angle  of  the  trian- 
gle A  0  C,  it  is  e- 
qual  to  CAO-\- 
ACO,i\mi'is,AOH 
is  equal  to  half  the 
sum  of  the  angles 
CAB,  BCA,  or  to  45°.     Produce  CO  to  II;  from  A  let  fall  All 


234  ROBINSON'S  SEQUEL. 

perpendicular  on  CA.      Now  in  the    A  A  OH,  because  ^=90°, 
and  ^0//=45°,   OAII=45'',  and  consequently  AJI=  Off. 

Put  AC=a,  AO=x,   OC=x+d,  Off  and  Aff,  each  equal  to 
y.     Then  Cff=x-\-i/+d. 

In  the  A  AffO,  we  have  2y2_^2  ^jj 

In  the  A  ^^C',  we  have 

(x+y+d)'-\.y^=.a^  (2) 

Expanding, 

x''-{-y^-{-d'+{2x-\-2d)?/-\-2dx+y''  =a^       (3) 

Substituting  the  value  of  y^  and  y  from  (1),  and 

2x^-{-(2x+2d)-^-+2dx=a''—d^. 

Or,  2x''+j2'x''-\-j2dx-{-2dv=a''^^\ 

Dividing  by  (2-[->/2),  and  we  have 
2  I  J       «^ — <^^ 
^         2+V2 

Whence,  ar  =  — ^±^/m4-^^ 

PROBLEM  21 

To  detei-mine  a  triangle,  having  given  the  base,  the  2^^fp€ndicular, 
and  the  difference  of  the  two  sides. 

(See  figure  to  Problem  19.) 
Let  ABC  be   the  A-     Put  BD=^x,  DC^y,  AC=z,  AB= 
z+d,  AD=a,  BC^h. 

By  the  conditions,  ar-|-y=6  (1) 

x^-\^^-=z'-\-2dz-{-d'  (2) 

y'+a'=z^^ (3) 

By  subtraction,  x^—y^=2dz-\-d^  (4) 

Factoring,         (^+y)  (^ — y)=d(2z-\-^) 
That  is,  b(x—y)=d{2z+d) 

From  this  we  have  the  proportion, 

b     :     (2z+d)     :    :     d     :     (x—y) 

This  proportion  is  the  following  rule  given  in  trigonometry,  viz : 

In  any  plane  triangle,  as  the  base  is  to  the  sum  of  the  sides,  so  ii 
the  difference  of  the  sides  to  the  difference  of  the  segments  of  the  base. 


GEOMETRY.  t$6 

We  return  to  the  solution.     From  ( 1 )  we  have 

rr=a — y,  whence  a;^ — y'^=:ar — 2ay. 
From  (3),     z=Jy^-\-a^.     These  values  put  in  (4),  give 
a^  — 2ay=2c?7y2+^+c?2 


Squaring,     (^a^—d^Y—Aa{a^—d''  )2/+4a-r  =^d''y''+^o''d^ 
Or,         (a2— ^a^a_4^(„2_^2  jy_|_4(g2__^2  yf^^a^'d- 

4^  2  ^2 

a^ — d^ — ^ay-\-^y^  ==-  - 

a  2 — <;- 

a  .  ,    '.     2  ,o     ,      4«2(/2  5«2J2_^4 


ar^2y=±:dj^ 
^    a 


Whence,  y=-=p-( I 

^     2^2\  a^—d^  / 


PROBLEM  22. 

To  determine  a  triangle,  having  given  the  base,  the  perpendicular, 
and  the  rectangle,  or  product  of  the  two  sides. 

(See  figure  to  Problem  19.) 

Let  ABGhQ  the  A.      Put  BD=x,  DC^y,  BC=b,  AD=za, 
and  the  recangle,  {AB)  (AC)=c. 

Now  in  the  right  angled  triangles,  ADB,  ADC,  we  have 


AB=Jx^--\-aK       AC^Jy^+a\ 

Whence, 

Ux-+a-){Jy^^+a^^)==c 

(0 

And, 

x+y=l 

(2) 

From  (1), 

x^y^+a^x'+y')+a'=c' 

(3) 

From  (2), 

x^-\-y''=b''—tcy 

(4) 

This  value  substituted  in  (3),  gives 

x^y^J^a^b^—'^a^xy-{-a^  =c* 
x^y^—^^xy+a^—c^—aH^ 


2A2 


xy-^a^  =  ±:Jc^—aH 
xy=a''±:Jc^—a^b^  (6) 


236 


ROBINSON'S  SEQUEL. 


From  equations  (2)  and  (5)  the  values  of  x  and  y  can  be  de- 
termined. 


PEOBLEM  23. 

To  determine  a  triangle,  having  given  the  length  of  the  three  lines 
dravmfrom  the  three  angles  to  the  middle  of  the  opposite  sides. 

Let  ABC  be  the  A. 
Bisect  the  sides  AB  in  D, 
AC  in  F,  CB'mE. 

Put  AE=^a,  BF=h,  CD 
=e,  AD=u,  AF=x,  BE 

Now  by  (th.  39,  b.  i), 
we  have, 

x^+b^=4u^-\-4g^ 


(1) 
(2) 

(3) 


By  addition,    a^-\-b^-\-c^=7(x'--\-y'^+u'') 
Whence,         ^a^^^-^-c'^^^ix^Jl^iy^+iu'- 
From  (1),  2^2^c2  =  4a;2-f4y2 


(4) 


By  subtraction,  ^a^-{'b^-\-c^)—u^—c^=4u^ 
Or,  4a2_[-452_j_4(.2_7^^2_7c2^28w2 

4a3_J_462_3c2=:35«<2 


By  inference 
And, 


t^-±J^' 

2+4*2- 

-3c2 

V             35 

^-±J4«- 

2+4c2- 

-362 

rt^ 

36 

V=r^J'^ 

2+462- 

-3a2 

36 


PROBLEM  24. 

In  a  triangle,  having  given  the  three  sides,  to  find  the  radius  of  the 
inscribed  circle. 


GEOMETRY. 


237 


Let  ABC  be  the  A. 
From  the  center  of  the  cir- 
cle 0,  let  fall  the  perpen- 
diculars OG,  OE,  OD, 
on  the  sides. 

These  perpendiculars  are 
all  eqiial,  and  each  equal 
to  the  radius  required. 

Let  the  side  opposite  to 
the  angle  A,  be  represent- 
ed by  a,  the  side  opposite  B  by  h,  and  opposite  C  by  c.     Put  OE, 
OD,  (fee.  equal  to  r. 

It  is  obvious  that  the  double  area  of  the  A  BOC  is  expressed 
by  ar  ;  the  double  area  of  A  OB  by  cr  ;  the  double  area  of  ^  0  C 
by  hr;  Therefore,  the  double  area  of  ABC  is  (^a-{-h-\-c)r. 

From  A  let  drop  a  perpendicular  on  BC,  and  call  it  x. 

Then  cw=  the  double  area  of  ABC.     Consequently, 
{^a-\-b-\-c)r=iax  ( 1 ) 

The  perpendicular  from  A  will  divide  the  base  BC  into  two 
segments,  one  of  which  is  Jc^ — x'\  the  other,  Jb'^ — x^,  and  the 
sum  of  these  is  a  ;  therefore, 


(2) 


'^=a^—2aJb^—x^J^b^—x'- 
^aJb^—x^=a^-]-b^—c^ 


J^'' 


-X^=z 


2a 


Whence, 

Or,  ar=^62 — m"^ 

This  value  of  a;  put  in  (1),  gives 


Whence, 


__ajb''—m^ 
a-\-b-\-c 


the  required  result. 


PROBLEM  25. 
To  determine  a  right  angled  triangle,  having  given  the  side  of  the 
inscribed  square,  and  the  radius  of  the  inscribed  circle. 


23^  ROBINSON'S  SEQUEL. 

Lei  O  be  the  center  of  a  cir- 
cle, OH  or  OL=ir,  the  given 
radius,  BE  or  ED^=a,  a  side  of 
the  given  square. 

BO'i^  the  diagonal  of  r-,  and 
BD  is  the  diagonal  of  «-,  and 
B  OD  is  one  continuous  line. 

The  point  D  of  the  given 
square  may  be  in  the  circle,  in 
that  case  the  hypotenuse  touch- 
es the  circle  and  the  square  in 
the  same  point,  and  that  point  is  the  middle  of  the  hypotenuse. 
If  the  point  D  is  not  on  the  circumference,  it  must  be  without,  as 
liere  represented. 

Draw  Dt  to  touch  the  circle   in  t,  and  that  line  produced  both 
ways  will  define  the  hypotenuse. 

^1(7  and  ^C  will  meet  if  produced,  and   ABC  \\\\\  be  the  tri- 
angle required. 

OB^J<Zr,   BD=j2a,  £>0=(a—r)j2,  J)V=(a—r)j2-^r, 

Now  as  i)  is  a  point  without  a  circle,  and  Dt  touching  it,  we 
have  by  (th.  18,  b.  iii),  {Dty-==DVxDU;  that  is, 

(i)/)2  =  [(a_r)72~r]  [(a— >-)72-fr ]=2a=— 4ar-fr2. 


Whence,  Dfz=jZa^ — 4ar-|-r2=:c. 

Because  A  is  a  point  without  a  circle,  and  AH,  At,  lines  drawn 
touching  the  circle,   AH=At,  (th.  18,  b.  iii,  scho.  2.) 

Observe  that  KH=a — r=d.     Put  AH,  At,  each  equal  to  x  ; 
then  in  the  A  AXB  we  have  AK=:x — d,  AD==x-\-c,  KD^c 

Whence,  (A'-|-(-)^=a--j-(.tr — d)-. 

x^  +2CX+C''  =a2  J^x'^^Stdx-^d* 


26+2^ 
Now  the  value  of  x  being  known,  AB  is  known,  and  all  the 
sides  of  the  A  AKD.     But  the  A  AKD  is  proportional  to  the 
triangle  ABC,  and  gives  us 

AK    :     KD     ',    i    AB     :    BC 


GEOMETRY. 


239 


The  first  three  terms  of  this  proportion  being  known,  the  last 
is  known,  and  the  triangle  is  fully  determined. 

PROBLEM  26. 

To  determine^  a  triangle  and  the  radius  of  the  inscribed  circle,  hav- 
ing given  the  lengths  of  three  lines  dravm  from  the  three  angles  to  the 
center  of  that  circle. 

Let  ABC 
be  the  A,  0 
the  center  of 
the  circle. 

Put^O= 
a,     OB=ic, 

OC=:h.    AO 

bisects      the 
angle  A. 

Produce 
AO  to  D, 
Then  because 
the  angle  A  is  bisected,   CD  :  DB  :  :  AC  :  AB. 

Put  AB=x,  AC=^y,  and  let  the  ratio  of  AB  to  BD  be  n\  then 
nx=BD  and  ny=^  CD. 

Now  as  the  angle  C  is  bisected  by  (70,  we  have 
AC    :     CD     :    :    AO     :     OB 

That  is,  y      :      ny      :    :       a       :      OD 

Whence,  OD—na. 

Because  AD  bisects  the  angle  A,  we  have,  (th.  20,  b.  iii), 


Also, 
And, 

From  (1), 


xy=^a^  {\'\-nY  -^-n'^xy 
nx^  =c^-\-na^ 
ny^=b^-\-na^ 


xy. 


a'^ilJ^ny  _a^(\^n) 


\—n 


1— w2 
The  product  of  (2)  and  (3),  gives 

n^x^y'^—{c^-\-na^)  {h^^rui^) 
Squaring  (4),  and  multiplying  the  result  by  n'^,  also  gives 


(1) 

(2) 
(3) 

(4) 
(5) 


»««y^ 


(6) 


240  ROBINSON'S  SEQUEL. 

Equating  (5)  and  (6),  gives 

This  equation  contains  only  one  unknown  quantity  n,  but  it 
rises  to  the  fourth  power — hence  this  problem  is  not  susceptible 
of  a  solution  from  this  notation  short  of  an  equation  of  the  fourth 
degree. 

In  cases  where  a,  b,  and  c  are  numerically  given,  the  solution 
may  be  possible  through  an  equation  of  the  second  or  third  degree. 

We  perceive  by  the  figure,  that  if  b=:c,  x  must  equal  y. 

PROBLEM  27. 

To  determine  a  right  angled  triangle^  having  given  the  hypotenuse 
and  the  radius  of  the  inscribed  circle 

Let  ABC  be  the 
A,  £^0  the  radius 
of  the  circle.  AU 
=AI)=  X,  CD= 
CF=zy.  Then  AB 
=.r+r.  BC=y-{-r. 

By  the  right  an- 
gled triangle, 

={x+yy         {\) 
x-\-y=a    (2) 
Reducing      (1), 
gives      xy=^rx-\-ry 
+r-. 

That  is,  xy=ar-^r^  (3) 

From  (2)  and  (3),  x  and  y  are  easily  found. 


In  numerical  problems,  great  advantage  can  be  taken  of  mul- 
tiple numbers,  the  same  as  we  have  shown  in  common  algebra. 
The  following  example  will  be  sufficient. 

The  sum  of  the  two  sides  of  a  2)lane  triangle  is  1 1 55,  the  perpen- 
dicular drawn  from  the  angle  included  by  these  sides  to  the  base,  is 


GEOMETRY. 


241 


I  I  \ 

HUB 


300  ;  the  difference  of  the  segments  of  the  hose  is  495.     WIicU  are  the 
lengths  of  the  three  sides?  Am.  945,  375,  780. 

Write  the  given  numbers  in  order,  thus,  300,  495,  1156.  Di- 
vide them  by  16,  and  their  relation  is  20,  33,  77. 

The  two  latter  numbers  have  a  common  factor  1 1 ,  which  call 
a.     Put  5=20. 

Then  the  three  given  lines  will  be  h,  3a,  and  7a. 

Let   CB=x,.  then    AC=7a — x. 
BD=y,  then  AD=y-\-3a.     CD=.h. 
In  the  right  angled  A  CDB,  we 
have  y2_|_j2^^2  (1) 

AD  Q  gives 

(y_[.3a)2+52==(7a— :r)2      (2) 
Expanding  (1)  and  subtracting  (2)  from  it,  gives 
6ay+9a2  ==49^,2 — i4«a; 
3ay=20a2 — '^ax 
Divide  by  a  and  write  h  in  the  place  of  20,  then/ 

Sy=ab — Ix 
Squaring,  9y^=a''b''—14abx+49x^ 

From  (1),        9y2=_962  +  9x^ 

By  subtraction,    0=:(a^-\-9)b^—l4abx-{'40x' 
Divide  by  &  (or  20),  then 

0=(a^-\'9)b—14ax-\-2x^ 
4a;2— 28aa;=— 2a2  b—l  8b 

Add  (49a2),  4x^-^2Qax+49a^=49a^—2an^l8b 

=9a2— .360=729 
By  evolution,  2x — 7a  =  ±27 

2a:=77±27=50,  or  104 
a:=25,  or  54  -» 

Here  25  is  the  number  that  corresponds  with  the  problem  ; 
therefore,  jB  (7=25- 15=375.     We  multiply  by  15,   because  we 
reduced  the  numbers  in  the  first  place  by  dividing  by  15. 
16 


242  ROBINSON'S  SEQUEL. 

SECTION   II. 

TRIOONOJWETRir. 

We  shall  here  attempt  to  show  the  most  practical  method  of 
finding  the  circumference  of  a  circle  to  radius  unity  ;  and  of 
finding  the  sines  and  cosines. 

The  trigonometrical  equations  that  we  may  call  into  immediate 
use,  are  the  following  : 

We  number  them  as  they  are  numbered  in  Robinson's  Trigo- 
nometry. 

sin.(a-|-5)=sin.a  cos.6-l-cos.a  sm.h  (7) 

sin.(a — J)=sin.acos.6 — cos.asin.6  (8) 

cos.(a-|-6)=cos.a  cos.6 — sin.a  sin.6  (9) 

cos.(a — J)=cos.a  cos.5-|-sin.a  sin.5         (10) 

sin.2a+cos.^a=l  (1) 

sin.2a=2sin.a  cos.a  (30) 

Or,  sin.a=2sin.^a  cos.^a 

By  problem  23,  book  iv.  of  Robinson's  Geometry,  we  learn  that 
if  we  divide  the  radius  into  extreme  and  mean  ratio,  and  take  the" 
greater  segment,  that  segment  will  be  the  chord  of  36°. 

Let  1  be  the  radius  of  a  circle,  and  x  the  greater  segment  re- 
quired ;  then 

1     :    X    :   :    X    :     1 — x 

Whence,  a:=—i-f  ^^5=0.6180340,  the  chord  of  36°  in  a 
circle  whose  radius  is  unity. 

We  learn  by  theorem  6,  book  v,  Robinson's  Geometry,  that 
when  c  represents  any  chord  of  a  circle,  and  x  a.  chord  of  one- 
third  of  that  arc,  the  following  equation  will  exist : 

x^ — 3x= — c. 
Put  c=0.6 18034000,  and  a  solution  of  the  equation  gives  the 
chord  of  12°.  Again,  put  c  equal  to  the  chord  of  12°,  and  an- 
other application  of  the  equation  will  give  the  chord  of  4°,  and 
thus  by  the  successive  application  of  this  equation,  we  have  found 
the  following  values : 


TRIGONOMETRY.  24» 

1.  The  chord  of  36°=0.6 18034000. 

2.  The  chord  of  12°=0.209056903. 

3.  The  chord  of    4°=0.069798981. 

4.  The  chord  of  80=0.023270628, 

By  theorem  4,  book  v,  we  learn  that  if  c  represent  the  chord 
of  any  arc,  the  chord  of  hcdf  that  arc  will  be  represented  by 


Having  the  chord  of  80'  the  preceding  expression  gives  us  the 
chord  of  40',  20',  and  10',  as  follows  : 

Chord  of  40'=0.01 16355131. 
Chord  of  20'=0,0058 177679. 
Chord  of  10'==0.0029088819. 

The  chord  of  10'  so  nearly  coincides  with  the  arc  of  10',  that 
for  all  practical  purposes,  we  may  consider  the  chord  and  arc 
the  same  ;  then  the  semicircumference  must  be  1080  times 
0.0029088819,  or  3.141592462.  A  more  exact  determination 
gives  3.141592653+  for  the  length  of  180'',  when  the  radius  is 
unity. 

The  chords  of  all  arcs  under  10'  cun  be  found  from  that  chord, 
directly/  hy  division. 

As  the  sine  of  an  arc  is  half  the  chord  of  double  the  arc, 
therefore,  we  can  have  the  natural  sine  of  18°  by  dividing  the 
chord  of  36°  by  2. 

Having  the  sine  of  any  arc,  we  can  find  its  cosine  by  the  fol- 
lowing equation  : 

cos.  «s=^l — sin.^a 

When  sin.^a  is  a  very  small  fraction,  as  it  is  for  all  arcs  under 
10',  then  ^1 — sin.^a  is  very  nearly  equal  to  (1 — ^sin*a). 

By  the  foregoing  we  find  the  following  sims  and  cosines  : 

sin.  r=.000^.908881  cos.  1'=.9999999576 

sin.  2'==  .00058 17762.  cos.  2'=. 9999998802 

sin.  3'=.0008726643  cos.  3'=.9999996692 

sin.  4'=:.001 1635524  cos.  4'=,9999993231 


244  ROBINSON'S  SEQUEL. 

sin.    5'=.001 4544405  cos.    5'=.9999989423 

sin.    6'=.0017453286  cos.    6'=.9999984769 

sin.    7'=.0020362167  cos.    7'=.9999979269 

sin.    8'=. 002327 1036  cos.    8  =.9999972926 

sin.    9=.0026179916  cos.    9'=.9999965731 

sin.  10'=.0029088789  cos.  10'=.9999957689 

sin.  20  =.0058177378  cos.  20'=. 9999830770 

sin,  30'=.0087265343  cos.  30'=.9999618877 

sin.  40'=.01 16352640  cos.  40'=.9999323090 

From  the  chords  of  4°,  12°,  and  36°,  we  readily  find 

sin.    2°=. 0348995000  cos.    2°=.9993908139 

sin.    6°=. 1045284515  cos.    6°=. 994521 8389 

sin.  18°=.309017000O  cos.  18°=  .951 06466 19 

Having  the  foregoing  sines  and  cosines,  we  can  find  the  sines 
and  cosines  of  certain  other  arcs  as  follows  : 

Put  2a=  to  any  arc  whose  sine  is  known,  then  we  can  obtain 
the  sines  and  cosines  of  the  half  of  2a,  or  a,  by  the  following 
general  equations : 

cos.^a+sin.^a=l  (1) 

2cos.  a  sin.  a  =  sin.  2a  ( 2) 

Now  if  we  suppose  2a=18°,  we  have  sin.  2a=.3090 170000  ; 
and  by  substituting  this  value  of  sin.  2a,  and  adding  and  subtract- 
ing the  equations,  we  shall  have 

cos.  2  a+2cos.a  sin.  a+sin.^  a==  1 .3090 1 70000  ( 3) 

and     cos.^a— 2cos.asin.a+sin.2a=0.6909830000  (4) 
By  extracting  the   square   root, 

cos.a-f-sin.a=l. 1441228508  (5) 

cos.a— sin.a=0.8312532699  (6) 
By  adding  (6)  and  (6),  and  dividing  by  2,  we  find 
cos.a=cos.9°=. 9876880603 


TRIGONOMElTlEtY:  246 

Subtracting  (6)  from  (5),  and  dividing  by  2,  gives 
sin.a=sin.9°=.  1 564342904 

If  we  put  2a=6°,  a  like  operation  will  give  the  cosine  and  sine 
of  3°.  If  we  put  2a=2°,  a  like  operation  will  give  the  cosine  and 
sine  of  1°,  and  so  on. 

Again,  we  must  not  overlook  the  fact  that  the  cosine  of  2°  is 
the  same  value  as  the  sine  of  88°  ;  therefore,  if  we  put  2a=88°, 
an  operation  like  the  preceding  will  give  us  the  cosine  and  sine  of 
44°.  Another  operation  will  give  us  the  cosine  and  sine  of  22°, 
and  still  another  of  11°,  and  so  on. 

If  we  require  the  cosine  and  sine  of  any  particular  arc  that  we 
cannot  arrive  at,  by  any  of  these  subdivisions,  we  may  apply  th§ 
following  equations : 

sin.  (a-(-6) =sin.a  cos.6-j-cos.a  sin.5         (7) 
sin. (a — &)=sin.acos.6 — cos.asin.6         (8) 

For  example,  if  a=6°  and  5=1°,  and  we  have  the  cosine  and 
sine  of  6°  and  1°  ;  then  (7)  will  give  us  the  sine  of  7°,  and  equa- 
tion (8)  will  give  us  the  sine  of  5°. 

Whence  by  equations  (1),  (2),  and  (7),  (8),  the  sine  and  co- 
sine of  every  degree  of  the  quadrant  can  be  obtained  without  the 
trouble  of  fractional  parts  of  degrees. 

But  there  is  a  better  method  to  continue  the  table  after  a  begin- 
ning has  been  made,  which  we  illustrate  by  the  following  example  : 

Suppose  we  have  the  sine  and  cosine  of  15°  and  16°,  and  also 
the  sine  and  cosine  of  each  of  the  small  arcs  from  zero  to  5°  or 
6°,  and  require  the  sine  and  cosine  of  17°  or  of  any  other  arc 
under  20°,  we  would  operate  as  follows : 

Let  the  arc  ^J9=15°,  ^i>=2°;  then 
^^=17°,  ^6^=17°,  i>6^=17°+16° 
=32°. 

Draw  the  chord  BD.  Now  because  an 
angle  at  the  circumference  is  measured 
by  half  its  subtended  arc,  therefore, 
the  angle  7ii?i)=16°.  The  chord  BD 
is  double  the  sine  of  1°;  and  it  is  obvi- 
ous that  we  have  BD  and  all  the  angles  of  the  small  right  an- 


246  ROBINSON'S  SEQUEL. 

gled  triangle  nBD  ;  and  if  we  compute  Bn,  and  add  it  to  DH, 
the  sine  of  15°,  we  shall  have  BJEy  the  sine  of  17° ;  and  nD  sub- 
tracted from  CH\  will  give  the  cosine  of  17°. 
The  computation  is  as  follows  : 

(We  use  the  logarithmic  sines  and  cosines,  diminishing  the  indices  by  10. 
to  correspond  with  radius  unity  in  the  table  of  natural  sines.) 

Log.  sine  1° —2.241855 

Log.  2 ,   .301030 

Log.  of  ^i> 2.542885 —2.542885 

sine  16°...  .—1.440338       cosine . .  .—1.982842 
»i>.... 009621     —3.983223    %JS  .033554— -2.525727 

,  Nat.  cos.  15°.... 96593   Nat.  sin.  15°    .258820 

Nat.  cos.  17°. .  ..95631    Nat.  sin.  17°    .292374 

Thus  we  can  go  on  and  compute  the  sine  and  cosine  of  19°. 

RemarJc.  When  the  triangle  nBD  is  taken  sufficienly  small,  the 
chord  BD  is  confounded  with  the  arc,  and  the  triangle  is  then 
called  the  differential  triangle,  and  figures  largely  in  the  differen- 
tial calculus ;  and  by  it  we  can  readily  compute  the  sine  and  cosine 
of  (16°  r),  (15°  2'),  &c.,  having  the  sine  and  cosine  of  the  de- 
gree, whatever  it  may  be. 

If  we  were  making  a  table  of  sines  and  cosines  for  every  min- 
ute of  the  quadrant,  it  would  require  too  much  labor  to  use  the 
foregoing  equations  for  every  minute,  we  would  use  them  for 
every  degree,  and  then  fill  up  the  sines  and  cosines  for  the  inter- 
mediate minutes  by 

INTERPOLATION. 

In  the  appendix  to  Robinson's  University  Algebra,  standard 
edition,  is  the  following  formula  for  inserting  any  intermediate 
term  of  a  series  : 

In  this  formula  a  is  the  first  term  of  a  series  consisting  of  a,  a^, 
^2,  ttg,  &c.,  terms,  b  is  the  first  term  of  the  first  difference,  c  is 
the  first  term  of  the  second  difference,  and  so  on.    The  interval 


TRIGONOMETRY. 


«47 


between  two  given  numbers  in  the  series  is  always  to  be  taken  as 
unity,  therefore,  %  is  a  fractional  part  of  that  unit. 
The  following  example  will  clearly  illustrate. 

1.  Given  the  sines  of  1°,  2°,  3°,  4°,  6°,  and  6°,  to  find  the  sines 
of  1°  12',  2°  12',  3°  12',  and  1°  24',  2°  24',  or  to  find  the  sine  of 
any  arc  between  1°  and  3°  by  interpolation. 


(*«) 

Ist  diff. 

Sddiflf. 

3d  diflf. 

sin.  1°=.0174524035 

(-H) 

{-^) 

{-d) 

sin.  2°=.0348995000 

.0174470965 

sin.  3°=.O523359508 

.0174364508 

106457 

sin.  4°=.0697664685 

.0174205177 

159331 

52874 

fiin.  5°=.0871557450 

.0173992765 

212412 

53081 

sin.  6°=.1045284515 

.0173727165 

265600 

53188 

To  interpolate  12',  we  must  put  n  of  the  formula  =  ||. 
2  n — 1  4  n — 2  6 


Whence, 


And, 


10 

2          n — 1 
%= —     n* : 

10 


10 


n'. 


1  n—\ 


10 
48 


2  100  2         3        1000 

The  products  will  be  positive  or  negative  according  to  the  rules 
of  multiplication.  For  the  sine  of  any  arc  between  1°  and  2°,  we 
take  the  first  line  of  the  column  under  a  for  the  first  term  of  the 
series,  and  the  first  line  of  the  column  under  h  for  the  first  dif- 
ference, and  so  on. 

To  find  the  sine  of  any  arc  between  2°  and  3°,  we  must  take 
the  second  line  of  the  column  under  a  for  the  first  term  of  the 
series,  and  the  second  line  of  the  column  under  h  for  the  first 
difference,  and  so  on. 

Whence,  the  following  equations  : 

sin.  1°  12'=.0174524034+T\(.0174470965)+^3.ir(  106457)— 
yf«~o(  52874)  =  .0209424308 

sin.  2°  12'=.0348995000+y2^(.0174364508)+yfxr(  159331)— 
tHttC  53081)=.0383878114 


248  ROBINSON'S  SEQUEL. 

sin.  3°  12'=.0523359508+y\(.0174206177)+yf^(.212437)— 
i-UTr(-63213)=.0558214986 

If  we  put  w=||=,V»  we  can  find  the  sine  of  1°  24',  2°  24', 
and  3°  24',  in  precisely  the  same  manner. 

In  short,  if  we  put  n  equal  any  number  of  times  gV»  we  can 
find  the  sine  of  the  degree  and  that  number  of  minutes,  but  it  is 
best  to  be  regular,  and  find  the  sines  to  1°  12',  1°  24',  1°  36', 
and  so  on,  and  then  interpolate  again  between  the  numbers  thus 
found. 

Little  attention  has  been  paid  to  this  subject  of  late,  because  the 
labor  when  once  done,  is  done  forever  ;  and  it  has  all  been  done 
in  the  preceding  age ;  our  object  has  been  to  present  a  systematic 
view  of  the  whole  matter,  and  show  the  student  that  the  task  of 
computing  a  trigonometrical  table  is  not  so  great  as  is  generally 


We  have  thus  far  computed  natural  sines  and  cosines,  but  we 
generally  use  logarithmic  sines  and  cosines. 

To  find  the  logarithmic  sine,  we  simply  take  the  logarithm  of  the 
natural  sine  from  a  table  of  the  logarithms  of  numbers,  increasing  the 
index  bg  10. 

After  a  few  logarithmic  sines  have  been  found  at  equal  inter- 
vals of  arc,  then  the  intermediate  logarithms  can  be  found  directly 
by  interpolation. 

To  make  a  table  of  logarithmic  sines  true  to  six  places  of  deci- 
mals, we  must  compute  with  at  least  eight  decimal  places  ;  and 
to  make  a  table  true  to  nine  places  of  decimals,  we  must  compute 
with  twelve  decimal  places. 

To  show  the  advantage  of  working  on  a  large  scale,  we  will 
require  the  log.  sines  of  1°,  2°,  3°,  4°,  5°,  and  6°,  true  to  nin£ 
places  of  decimals. 

The  natural  sines  we  already  have,  and  the  necessary  tabl-es 
of  logarithms  are  in  the  latter  part  of  this  volume,  the  same  as  are 
to  be  found  in  our  Surveying  and  Navigation. 

Most  operators  would  take  out  the  logarithm  of  each  natural  sine 
separately,  having  no  connection  mth  eojch  other,  but  this  would  re- 
quire much  unnecessary  labor,  and  it  is  to  explain  the  artifices,  that 
we  bring  forward  the  example. 


TRIGONOMETRY.  249 

In  the  first  place  we  will  take  the  sine  of  6°,  that  is,  find  the 
log.  of  the  number  .1045284515. 

Log.  .104 —1.017033339299 

Factor,  1.005 Table  A, .002166071750 

Log.  prod.,  .  104520^7 01919941 1049 

1.00008.. .  Table  C, 34742166 

.104528  36160         W 019234153215 

o 

Dividing  the  given  number  by  this  number,  we  find  another 
factor  to  be  1.0000008.  The  log.  of  this  factor  corresponds  to 
8(c)  in  table  C  ;  therefore, 

—1.019234153215 
347432 

Log.  .1045284515= —1.019234500647,  nearly. 

Add  10. 

Tabular  log.  sine  6°=  9.019234500647 

Having  the  logarithm  of  .1045284515,  and  requiring  that  of 
.0871557425,  we  first  consider  whether  we  cannot  find  some  con- 
venient divisor  to  the  first  that  will  produce  the  second  for  a 
quotient,  or  produce  a  number  very  near  the  second.  To  find 
definitely  what  this  divisor  is,  represent  it  by  D  ;  then 

•^^^^^-=0.087155.     Whence,  i>=1.2,  nearly. 

Log.  .1045284515 —1.019234500647 

Divide  by  1.2)  log.       0.079181246048 

Gives  .087107043  log.  —2.940053254599 


Factors, 


1.0005 
1.00005 
1.000008 
1.0000008 


217099966 

Table  C, 21714178 

3474352 

347435 


Prod,  nearly  =  .0871657425     log.  nearly=— 2.940295890530 

We  found  these  factors  by  taking  .0871557425  for  a  dividend, 
and  .087107043  for  a  divisor;  the  quotient  is  1.0005588,  which 


260  ROBINSON'S  SEQUEL. 

we  directly  separate  into  the  single  factors,  1 .0005,  1 .00006,  <fec. 
(See  Art.  14,  Robinson's  Surveying  and  Navigation.) 

We  can  easily  find  the  logarithmic   sine  of  1°,  because  the 
number  .0174524035  happens  to  be  peculiarly  favorable  ;  174= 
87-2,   and   174  multiplied  by  1003,  gives   174622.      Whence, 
.87X.02X(1.003)=.0174622. 
.0174524036 


Again, 


'=1.00001166. 


Factors, 


.017422 

'  Log.    .02 
Log.    .87 
Log.  1.003 
Log.  1.00001 
Log.  1.000001 
Locr,  1.0000006 


. . . .  —2.301  029  995  644 
....—1.939  519  252  619 

001300  943  017 

Tabled  4  342  923(a) 

434  294(5) 

260  574  (6c) 

26  058(6rf) 


Log.  1.00000006 

Product  nearly     .0174524035     log.  —2.241  855  255129 

Add  10. 
Tabular  log.  of  1°,  therefore,  is 


8.241  865  255129 

To  find  the  logarithmic  sine  of  2°,  we  proceed  thus  :  To  the 
log.  of  the  sine  of  1°,  add  the  log.  of  the  number  2,  then  we  shall 
have  the  logarithm  of  a  number  a  little  above  the  one  required, 
which  can  be  reduced  by  division. 

Log.  .0174524035 —2.241855255129 

Add  log.  2, .301029995644 

Log.  .034904807 —2.642885250773 

Divide  by  1.000152     Sub.  log.  1.0001  43427277 

—2.642841822496 
•    Alsosub.log.  1.00005 21714178 

—2.642820108318 

Also  sub.  log.  1.000002 .^ 868588  (2b) 

Log.  .0348996= —2.542819239730 very 

Add     10.  nearly. 

Tabular  log.  3°  (true  to  10  places),  =  . .  .8.5428192397 


TRIGONOMETRY.  261 

We  found  the  divisor  1.000152  by  the  following  equation. 
Calling  D  the  divisor  sought,  then 

.034904807^  Qg^ggg^     Whence,  i>=1.000152. 

By  a  little  examination,  we  shall  find  that  if  we  multiply  the 
sine  of  1°  by  3,  and  divide  the  product  by  1.000406,  the  quotient 
will  be  the  sine  of  3°  very  nearly. 

To  the  log. —2.241  856  255  129 

Add  log.  3, 477  121  254  720 

—2.718  976  509  849 

Sub.  log  1.0004 '.^ 173  690  053 

—2.718  802  819  796 
Also,  sub.  log.  1 .000006 2  605  764  (26) 

Log.  sine  of  3°= —2.718  800  214032 

AddJlO^ 

Tabular  log.  sine  3°= 8.718  800  214  032 

In  a  similar  manner  we  can  find  the  logarithmic  sine  of  4°. 

If  it  were  our  object  to  compute  a  table  of  logarithmic  sines 
and  cosines  for  every  degree  and  minute  of  the  quadrant,  we 
would  first  compute  each  degree  and  half  degree  in  natural  num- 
bers— and  take  the  logarithms  of  those  numbers.  Then  we  would 
interpolate  for  the  intermediate  logarithms. 


We  now  proceed  to  solve  proUeTns  m  Trigonometry  and  Menswrc- 

tion. 

(Problems  1  and  2  are  on  page  167,  Robinson's  Geometry.) 

1.   Given  AB  428,  the  angle  C  49°  16',  and  (AC+CB),  918,  to 
find  the  other  parts. 

Let  ABC  represent 
the  A.  Draw^I>=918. 
From  D  draw  DB  so 
thatthe  angle  ADB  shall 
be  half  the  angle  ACB, 
that  is,  24°  38'.  From 
^  as  a  center  with  AB 
as  a  radius,  strike  an  arc 


252  ROBINSON'S  SEQUEL. 

cutting  BDm  B.     From  B  make  the  angle  I)BC=2^°  38': 

then  A  CB  will  be  the  A  required. 
In  the  AADB  we  have  AB  :  AD  :  :  sin.  D  :  sin.  ABD, 
That  is,         428     :     918     :    :     sin.  24°  38'     :     sin.  ^^i>. 

Sin.  24°  38' 9.619938 

Log.  918 2.962843 

12.682781 

Log.  428 2.631444 

Sin.  63°  22'  48"  or  its  supplement  116°  37'  12".  .9.951337 

From  this  take  DB  C 24°  38' 

ABC= 9r°^59M2" 

Having  now  two  angles  of  the  A  ABC,  we  have  the  third  angle 
^=38°  44'  48",  and  with  all  the  angles,  and  the  side  AB,  we 
find  A (7=564.49,  and  conseqently  5(7=354.51. 

(2.)  Given  a  side  and  its  opposite  angle,  and  the  difference  of  the 
other  two  sides,  to  construct  the  triangle  and  find  the  other  parts. 

Let  ABC  be  the  A.  ^(7=126,  5= 
29°  46',  and  AM,  the  diflference  between 
AB  and  BC,  =43. 

From  180°  take  29°  46'  and  divide  the 
remainder  by  2.  This  gives  the  angle 
BMC  or  BCM.  BMC  taken  from  180°, 
gives  AMC. 

Now  in  the   A  AMC,  we  have  the  two  sides  AC,  126,  AM, 

43,  and  the  angle  AMC,  to  find  the  angle  A.      The  computation 

is  as  follows  :    180°— 29°  46'=150°  14'  ;    half,  =75°  r=BMC. 

180°— 75°  7'=  104°  52,'= AMC.     Now  in  the  A  AMC,  we  have 

AC    :     AM    '.    :     sin.  104°  63'     :     &m.  ACM 

Sin.  104°  63'=cos.  14°  63'. 

126     :     43     :    :     cos.  14°  53'     :     sm.  ACM 

Cos.  14°  63'. 9.986180 

Log.  43 1.633468 

11.618648 

Log.  126 2.100371 

fiin.  ^C7Jf=sin.  19°  22'  28" 9.518277 


i 


TRIGONOMETRY. 


253 


Whence,  ^(7^=76°  7'+ 19°  22'  28"=94°  29'  28".  Conse- 
quently ^=55°  51'  32".  Now  we  have  all  the  angles,  and  AC, 
of  the  A  ABC. 

(3.)  Two  lines  meet,  making  an  angle  of  50°.  On  one  line  are 
two  objects,  one  200,  the  other  500  yards  from  the  angular  point. 
Where  abouts  on  the  other  line  will  these  two  objects  appear  under  the 
greatest  possible  angle,  and  what  unll  that  angle  be? 

LetP^andPi) 
be  the  two  lines, 
and  A  and  B  the 
two  objects. 

Let  i)  be  the  re- 
quired point  on  the 
other  line.      Then 

pd=jTaxpb 


=  J500X  200  = 
316.226+  yards; 
but  this  requires 
demonstration. 

If  we  make  PD=JPAxPB,  and  then  pass  a  circle  through 
the  points  A,  B,  and  D,  PD  will  touch  the  circle  in  the  point 
D.  (Th.  18,  b.  3,  scho.)  And  because  PD  is  a  tangent,  the 
angle  ABB  at  the  point  of  contact,  is  greater  than  any  other  an- 
gle AdB,  on  either  side  of  D,  (see  th.  7,  page  101  of  this  volume.) 
Or  we  may  prove  it  here.  AeB=ADB,  (th.  9,  b.  iii,  scho.); 
but  AeB  is  greater  than  AdB,  therefore,  ADB  is  greater  than 
AdB  ;  that  is,  greater  than  any  angle  drawn  from  any  point  be- 
tween P  and  D.  The  same  demonstration  will  apply  on  the  other 
side  of  D. 

The  computation  for  the  angle,  is  as  follows  : 

From  D  let  drop  the  perpendicular  DH,  then  in  the  A  PDH, 
we  have 

As  radius,         10.000000  10.000000  ^ 

To  PD.  2.600000  2.500000  * 


So  is  sine  50°, 
To  DH,  242.24, 


9.884262 


cosme 


9.808067 
2.384252    PH,  203.26,    2.308067 

m 


254 


ROBINSON'S  SEQUEL. 


From  PH  take  PB,  and  we  have  JIB==3.26.     From  PA  take 
PJI,  and  we  have  Aff=^296.S4. 
Now 


HB     \    HD    \    \ 

;     R    : 

:    tang.  ABD, 

AH    '.    HD     '.    \ 

:    E     ; 

:     tang.  BAD. 

12.384252. 

12.384252 

HB        0,513218 

AH        2.471800 

Tan.  ABB  89°  14'  11.871034   tan.  BAB  39°  16'     9.912462 

180°— (89°  14'+39°  16') =51°  SO'=:ABB,  the  greatest  angle 
required. 

At  the  point  G  on  the  line  PG,  the  objects  A  and  B  would 
extend  the  greatest  possible  angle,  and  in  that  case  also,  PG= 
JPAxPB7~ But  the  angle  A GB  must  be  of  such  a  value  that 
AI)B+AGs=nO'' ;  therefore,  ^6^^=128°  30'. 

(The  following  are  on  page  174,  Robinson's  Geometry.) 
(3.)  From  an  eminence  q/*  268  feet  in  perpendicular  height,  ike 
angle  of  depression  of  the  top  of  a  steeple  which  stood  on  the  same 
horizontal  plane,  was  found  to  he  40°  3',  and  of  the  bottom  56°  1 8'. 
What  was  the  height  of  the  steeple?  Ans.  1 1 7.8  feet. 

Let  BC  hQ  the  eminence  268  feet, 
and  AD  the  steeple.  Draw  CE  par- 
allel to  the  horizontal  AB.  Then 
JSCD=40''  3',  £CA=CAB=56°  18'. 
i)(7^=56°  18'— 40°  3'=16°  15'. 
2)^C=90°— 66°  18'=33°  42' 

In  the  A  ABO,  we  have 
sin.  Qe""  18'  :  268  :  :  sin.  90°  :  AC. 
^^_    268X-g 
sin.  56°  18' 
In  the  A  ADC,  we  have  the  supplement  to  the  angle  ADC 
equal  to  16°  15'  added  to  33°  42',  or  49°  57' ;  therefore. 
As  sin.  ADC    :    AC    :    :    sm.  DCA     :    AD 

268Xi2 


That  is,    sin.  49°  67'  : 


sin.  56°  18' 


sin.  16°  15'  :  ^i> 


TRIGONOMETRY.  256 

^7)^     268'.a-sm.  16°  15^     _2.428135+10.+9.446893  __ 
sin.  49°  67' -sin.  66°  18'  9.883836+9.920099" ~" 

21.876028—19.803936=2.071093 
Log.  ^i>=2.071093.   Whence,  ^2>=1 17.78  feet. 

(4.)  From,  the  top  of  a  mountain  three  miles  in  height,  the  visible 
horizon  appeared  depressed  2°  13'  27".  Required  the  diameter  of 
the  earth,  and  the  distance  of  the  boundary  of  the  visible  horizon. 

Ans.  Diameter  of  the  earth  7968  miles,  distance  of  the  hori- 
zon 164.64  miles. 

Let  AB  represent  the  mountain,  and 
AD  the  visible  distance.  AB  produced 
will  pass  through  the  center  of  the  earth 
at  C.  From  D  draw  CD  perpendicular 
to  AD.  Join  BD.  AD  (7  is  a  right  an- 
gled triangle. 

C^i>=90°— 2°  13'  27"=87°  46'  33". 
ACD=9P  13'  27".  ADB=^ACD= 
r  6'  44".     ^^i>=91°  6'  44".  " 

No-jF  in  the  A  ABD,  we  have 

sin.  1°  6'  44"     :     3     :    :     sin.  91°  6'  44"     :     AD. 

Sin.  91°  6'  44"=cos.  1°  6'  44" 9.999919 

Log  3 0.477121 

10.477030 

Sin.   1°  6'  44" 8.288029 

Log.  164.64 2.189001 

in  the  triangle  AD  C,  we  have 

sm.ACD     :    AD     :   :    cos.  A  CD    :     CD 

Cos.  -4Ci>=cos.  2°  13'  27" 9.999674 

AD ^ 2.189001 

12.188676 
Sin.  ACD=sm.  2°  13'  27" 8.588932 

3.699743 

Double 0.301030 

Diameter log.  7958  miles,  nearly 3.900773 


256  ROBINSON'S  SEQUEL. 

(Several  of  tlie  following  problems  in  Mensuration  are  taken  from  the 
Surveying  and  Navigation,  page  60.) 

(5.)  Find  the  length  of  an  arc  of  30°,  the  radius  being  9. 

When  the  radius  is  1,  an  arc  of  180°=3. 141592  ;  therefore, 

3  141592 
an  arc  of  30°  and  radius  1  must  be  — ,  and  this  multiplied 

by  9  must  be  the  required  result. 

„              3.141592-3     .  „,c>«oo     A 
Hence,     =4.712388,  Ans. 


(6.)  Find  the  area  of  a  circular  sector  whose  arc  is  18°,  and  ra- 
dius 1^. 

We  must  first  find  the  length  of  the  arc,  as  in  the  last  problem, 
then  multiply  its  half  by  the  radius. 

•^  141  fiQ9 

Whence,       _LlZ:!^ixl8X  ^=.3141592Xf=.235619=-»  arc. 
180 

Therefore  the  area  must  be  fX 0-2356 19=0.363427,  Ans. 

The  arc  of  1°  and  radius  unity  is  .0174533. 

Therefore,  that  of  9°  is  .0174533X9,  and  this  multiplied  by 

the  square  of  the  radius  will  give  tlie  true  result. 

That  is,   .0174533X9X1=0.353403. 

(7.)  Required  the  area  of  a  sector  whose  radius  is  26,  and  ar- 
147°  29'. 

.0174533Xl47.4833x625__g^^  3^^^ 
2 

(8.)  What  is  the  length  of  a  chord  which  cuts  off  one-third  of  the 
area  from  a  circle  whose  diameter  is  289  ?  Ans.  278.6716. 

Like  many  problems  in  relation 
to  the  circle,  this  can  be  solved  only  by 
approximation . 

As  circles  are  in  all  respects  pro- 
portional to  their  radii,  I  will  ope- 
rate on  radius  unity,  and  in  conclu- 
sion, multiply  by  2-|^, 

If  the  segment  FFD=}  of  the 
whole  circle,  ABDE  will  equal  |  of 
the  whole.     Because  ^ — |=i. 


TRIGONOMETRY.  267 

The  space  ABDE  contains  two  equal  sectors,  DCB,  ACE^ 
and  the  triande  ^Ci).      Put  the  arc  BD=x,  CB^\.      Then 

o 

C6^=sin.  X,  OD=cos.  x. 
The  area  of  the  two  sectors  together  is  x. 

The  area  of  the  triangle  ECD  is     sin.  x  cos.  x. 

Therefore,         ir+sin.  x  cos.  x=\7i.  rt=3. 141 592653+ 

Double,  2a;-|-2sin.  x  cos.  x=^7t. 

But  sin.  2.2;=2sin.  x  cos.  x.     (See  eq.  (30),  page  143,  Geom.) 

Therefore,  2ar-[-sin.  2x^^7t  (1) 

Here  we  have  a  correct  and  definite  equation,  but  we  cannot 

solve  it,  as  it  contains  an  arc  and  its  sine,  and  they  are  not  united. 

by  any  definite  numerical  law  ;   we  must,   therefore,   resort  to 

ajyproximation. 

We  know  that  sin.  2x  is  not  much  less  than  2x. 
Therefore,     4x=^7i  is  not  far  from  the  truth. 
Also,     2  sin.  2x=}7t  is  not  far  from  the  truth. — The  one  too 
small,  the  other  too  large. 

That  is,  x=z^^rt,  approximately,  and  sin.  2a:=i7t,  approximately. 

To  find  the  arc  BD  approximately,  we  have  this  proportion : 

rt     :     j\7t  :  :    180°     :         Arc  BD.     Whence,  ^i>=  15°. 

By   the  table  of  natural  sines  we  find  sin.  2ii;  =  sin.  31°  34' 
nearly.     Or,  ic=15°  47'  nearly. 

}Ve  nozv  knoio  that  to  make  the  area  ABDE=}7t,  the  arc  BD 
must  be  greater  than  15°  and  less  than  15°  47'. 

I  will  now  suppose  the  arc  BD=15°  20',  and  compute  the  area 
ABDE,  corresponding  to  that  supposition. 

For  the  numerical  value  of  the  arc  15°  20',  we  have  the  follow- 
nig  proportion  : 

180°     :     15}j°     :    :     3.14159265     :     Mq  BD 

Or,   540     :     46     :    :     3.14159265     :     Arc^i)=0.2676175. 

The  tables  will  give  us  (sin.   15°  20')  (cos.  J5°  20')  thus  : 
17 


258  ROBINSON'S  SEQUEL. 

Sin.  15°  20' 9.422318 

Cos.  15°  20' 9.984259 

Sum  less  20=log — 1.406577=0.2550200  nearly. 

jBi>=a;=0.2676175  nearly. 

Area  ABDE= 0.5226375  nearly. 

But  the  required  area  of  ABDE  is  \7t— 0.5235987  nearly. 

Hence,  15°  20' for  ^i>,  gives  an  area  too  small  by  0.00096 12 

Now  we  wish  to  increase  the  area 

ABDE  by  the  little   narrow  space 

EDdey  and  this  is  so  narrow  that 

Dd  and  Ee  are  in  respect  to  practi- 
cal or  numerical  purposes,  right  lines, 

and  EDcle  is  a  trapezoid,  and  its  par- 

alel  sides  may  be  taken  as  equal ;  it 

is  then  practically  a    parallelogram 

whose  area  is  given  and  its  longer  side 

equal  to  2(cos.  15°  20'). 

Let  y=  the  width  of  this  parallelogram  or  trapezoid,  (as  we 

may  call  it  either.)     Then  we  shall  have  the  following  equation  : 
2cos.(15°  20')y=0.0009612 
Or,  cos.(15°  20')y=0.0004806 

That  is,  0.9644^=0.0004806.  Whence,  y=0.000498 

That  is,  we  must  increase  the  natural  sine  of  BJ)  15°  20',  by 

0.000498. 

The  natural  sine  of  15°  20'  is 0.264434 

To  which  add 0.000498 

N.  sin.  of  15°  21'  47"  cor.  to  sum 0.264932 

Thus  we  learn  that  the  arc  Bd  corresponds  to  15°  21'  47"  as 

nearly  as  a  table  of  natural  sines  computed  to  6  decimal  places 

will  give  it. 

Twice  the  cosine  of  16°  21'  47",  to  a  radius  of  ^(289)  is  the 

chord  sought,  which  we  compute  as  follows  : 

Cos.  15°  21'  47" 9.984184 

Log.  289 2.460898 

Log.  278.67+ 2.445082 


TRIGONOMETRY. 


269 


(9.)  WTiat  is  the  radius  of  a  circle  whose  center  being  taken  in  the 
circumference  of  another  containing  an  acre,  sliall  ciU  of  half  of  its 
contents  /*  , 

This  problem  is  the  same  for  circles  of  every  magnitude ; 
therefore,  we  will  operate  on  a  circle  of  radius  unitt/. 

Let  X  represent  the  number 
of  degrees  in  the  arc  AB,  and 


180 


the  length  of  each  degree; 


TCX 


then  "'*'  represents  the  length 
180     ^  ^ 

of  the  arc  AB. 

BF=  sin.  X.  CF=  cos.  x. 
FA=\  —  cQs.x.  (ABy  = 
(1 — cos.a;)2-|-sin.2^,  or  AB 
=  jj2 — 2cos.iC,  which  equals 
the  radius  of  the  cutting  circle. 

The    area   of  the    sector  CBAD^    is  measured  by    the  arc 

AB'CA;  that  is,  ---,'  From  this  take  the  triangle  CBD,  or 
180  6 

sin.  a:  cos.  a;,  and  the  segment  ABB  will  be  left.     That  is, 

Segment  ABFJ)=—— sin.  x  cos.  x.     (^=3.141592.) 

*Oiie  reason  for  the  appearance  of  this  work  is  that  it  is  required,  because 
able  mathematicians  have  written  so  obscurely.  They  seem  to  have  written 
as  I  should,  were  I  indifferent  whether  the  reader,  or  rather  the  learner,  un- 
derstood me  or  not.  Do  not  the  following  extracted  solutions  justify  this 
observation  ?  they  are  brief,  to  be  sure,  and  no  one  sets  a  higher  value  on 
brevity  than  does  the  author  of  this  work  ; — but  nothing  is  meritorious 
which  is  ^^anting  in  perspicuity. 

The  following  extracts  are  from  the  Mathematical  Diary,  published  by 
James  Ryan,  1825. 

Solution. — By  Robert  Adrain,  LL.  D. 

In  a  circle  to  radius  unity,  let  2«  be  the  arc  of  which  the  chord  is  the  re- 
quired radius,  then  ?r  being  the  area  of  the  given  circle  to  radius  unity,  if  we 
express  analytically  the  area  cut  off  by  the  radius  sought  and  divide  by  2, 
we  obtain  the  transcendental  equation 


(^^z\cos.  22+1  sin.  2^=1 


260  ROBINSON'S  SEQUEL. 

Again,  as  x=  tlie  degrees  in  AB,  (180 — x)  =  the  degrees 
in  BE.  Because  the  angle  BAH  is  at  the  circumference,  it  is 
measured  by  \  of  (180— a;),  or  (90— iic^. 

Whence,  the  arc  BJI=90 —  ^  x,  measured  in  degrees. 

For  the  length  of  the  arc  BH,  we  observe  that  180°  of  the  cir- 
cumference would  be  measured  by  7t^2 — 2cos.a?. 

for  1    then,  we  have  -^ _1_,  this  multiplied  by  the  num- 
ber of  degrees,  (90-|)will  produce  (?.?/?^|^V90-?) 

for  the  linear  measure  of  the  arc  BIT. 

This  multiplied  by  the  radius  AB,  or  J2 — 2cos.ar,  will  give  the 
area  of  the  sector  ABHD  ;  that  is, 

Sector  ABHD=^(^-i^^y\Uo-'^\ 

From  this  subtract  the  triangle  ABD,  which  is  measured  by 
sin.  a:(l — cos.a:),  and  we  have  the  segment  BIIDF. 
That  is. 

Segment  ^^i>^=<l:^e!:^(i??:^^^^sin.:r+sin..;  cos.;r. 
90  \      2      /  ' 

But  segment  ABFD=i  —- — sin.  x  cos.  x. 
^  180 

The  sum  of  these  two  segments  is  the  double  circular  space, 

ABHD  required  ;  that  is, 

Hence,  «=35°  24',  and  therefore,  if  R=the  radius  of  the  given  circle,  the 
radius  sought  =  2r  sin.  (35°24')=1.158e. 

Solution. — By  Dr.  Henry  J.  Anderson. 
Let  the  radius  of  the  given  circle  be  represented  by  unity,  and  of  the  two 
portions  of  its  circumference  terminated  at  the  intersections  of  the  two  circles 
let  the  greater  be  denoted  by  2<}>.  Then,  by  the  rules  of  mensuration,  it  will 
be  found  that  the  two  parts  into  which  the  given  circle  is  divided  are 
equal,  each  to  2<j)  coe,^  i<p-^7r — <p — sin.  <p.  Putting  this  equal  iv,  the  area 
of  the  semicircle,  and  transposing,  we  have 

sin.  <t> — (2cos.^i4) — 1)=-,     Or  by  trigo.,  sin.^ — ^cos.4)=-, 

whence,  ^=109°  11'  17"  and  the  required  radius =1.15874. 

If  the  contents  of  the  given  circle  be  one  acre,  then  the  required  radius  will 
be  206.7336  links,  or  about  45.4814  yards. 


TRIGONOMETRY.  261 

The  area  ABHD=^—^m.x-\-—J(\—cos.x)(\^0.^-x)\ 
180  '  180\^  ^^  ^/ 

This  reduces  to  (  1— cos.  x-\- \  — sin.  x. 

If  we  put  this  expression  equal  to  the  given  quantity,  _  we 

jit 

cannot  resolve  the  equation,  because  it  would  contain  the  linear 
quantity  x  and  the  transcendental  quantities  sin.  x  and  cos.  x. 
Therefore  if  we  solve  the  problem  at  all,  we  must  do  it  indirectly, 
by  approximation,  as  we  are  obliged  to  do  with  nearly  all 
problems  pertaining  to  the  circle. 

This  expression  is  a  general  one,  and  if  we  assume  x  any  num- 
ber of  degrees,  we  can  readily  obtain  the  corresponding  value  of 
the  expression,  and  if  any  assumption  corresponds  to  a  given  value, 
the  problem  is  solved  ;  and  if  it  nearly  corresponds,  we  shall 
have  nearly  the  radius  required,  which  can  be  increased  or  de- 
creased, as  we  are  about  to  explain. 

For  the  area  ABHD  to  contain  half  of  the  circle,  it  is  our 
judgment  that  the  arc  AB  should  contain  about  76°  ;  therefore, 
we  assume  x=-lb^  and  the  expression  becomes 

(^              _P  ,    10  COS.75\7t        .     ^e 
1 — COS.754- I  — sin.75. 
^       24       / 

By  the  table  of  natural  sine^  we  find 

(0.741 18+0.10787)7t— 0-96693 

The  final  result  of  this  supposition  is  that  the  area  ABHD=^ 
1.701370.  But  the  half  of  the  circle  is  1.570796  ;  therefore,  we 
have  taken  x  too  great  to  obtain  the  area  of  half  the  circle. 

We  will  now  take  x=10°. 

Then  A— cos.70+!^^^V— .sin.70°  will  be  an  expression 
for  a  less  area  than  before. 

By  log.  log.  0.79099 —1.898175 

log.  3.1415926 0.497149 

2.48500 0.395324 

Nat.  sin.  70^ 93969 

Area  ABHD 1.54531 

Given  result .1.57079 

Error  too  small 0.02548 


t62  ROBINSON'S  SEQUEL. 

This  error  must  be  conceived  to  be  a  winding  'parallelogram, 
whose  length  is  BHD.  Dividing  .02548  by  BHBy  will  give  the 
amount  to  be  added  to  the  radius  AH.  The  radius  AH  or  AB 
is  2sin.35°=l. 14716. 

The  angle  ^^li)=180— 70=1 10°. 

The  Unear  value  of  ^^i>=Qif!im^^i!il^??)li^. 

180 

Now  the  amount  that  the  radius  must  be  increased  is  expressed 

^^  (.02548)18 


(1.14716)  (3.141592)11 

By  logarithms.     Log.  .02548 —2.406199 

Log.         18 1.255273 

—1.661472 
Log.  1.14716 0.0596187 

Log.  3.141592 0.4971499 

Log.  11 1.0413927 

1.5981613 1.698161 


0.01 1 568 —2.06331 1 

Add... 1.14716 

1.158728=  the  required  jradius  AB,  which  will  cut  the 
circle  into  two  equal  parts. 

If  the  radius  of  the  given  circle  is  (a),  in  place  of  unity,  then 
the  radius  of  the  cutting  circle  must  be 
(1.15828)a 
To  find  the  number  «>f  degrees  and  minutes  m  ABy  divide 
1.15828  by  2,  which  gives  .57914  for  the  sine  of  half  AB,  or 
35°  23'  30"  or  ^jB=70°  47'. 


The  following  theorems  are  extracted  from  pages  219  and  220 
of  Robinson's  Geometry, 

(1.)  Show  geometrically,  that  R(R-l-.cos.  A)  =  2  cos.' ^  A ;  and 
that  R(R— cos.  A)=2  sin.'^  A. 


TRIGONOMETRY.  S^5 

Let  CB  or  CA  represent  the  radius  of 
a  circle  and  call  it  E.  Let  the  arc  ^i>= 
Ay  and  draw  the  lines  here  represented. 

Then  GD=^m.A,  CG=cos.A,  BG= 
E-]-Qos.A,  OA—R — cos.^,  AI=^m.\A, 
CI=co8.\A,  jBi>=2cos.J^. 

From    C  draw   CO  perpendicular  to 
JBD  ;  then  JB  0=  OD,  and  the  two  A's 
BOCy  Bl) G,  are  equiangular  ;  therefore, 

BG    :    BD    :   :    BO     :    BC. 

That  is,       jB-j-cos.^  :  2cos.  ^-4  :  :  cos.^J^  :  B. 
Whence,        B(B+cos.A)=2cos.^  I  A.  Q.  E.  D. 

Again,  by  the  similar  A's  AGDy  ADB,  we  have 

AG    :    AD     :    :    AD     :    AB, 

That  is,         i2— cos.  A  :  2sin.  ^A  :  :  2sin.i^  :  2E. 
Whence,       B(E—cos.  A)=2sm.^A.  Q,  E.  D. 

(2.)  Show  that  R'sin.  A=2sin.^  A  cos.^  A. 
By  similar  triangles,  we  have 

AG    I     CI    :   :    AD    :    DG. 

That  is,     E     :     cos.  ^A     :    :     2sin.  ^A     :     sin.  A. 
Whence,         E  s'm.  A=2sm.^Acos.^A.         Q.  E.  D. 

(3.)  Prove  that  tan.  A-}-tan.  B=  ^  "^ — ^~t — I  radius beinff  unity. 

COS.  A  COS.  B 

It  is  admitted  that  tan.^= — '- — ,  and  tan.  Bz 


By  addition,    tan.  A-{-tsLn.  B= 


COS.Jl  COS.  ^ 

sin  A  ,  sin.B 


COS. A     cos.B 


sin,  A  cos.  ^-f-cos.  A  sin.  B sm.(A-\-B)      n    E    D 

COS. -4  COS.  ^  cos.-4cos.  ^ 

(4.)  Demonstrate  geometrically y  that  R  sec.  2 A  =  tan.  A  tan.  2 A 
+R2. 


264 


ROBINSON'S  SEQUEL. 


Take  CB  radius,  let  the  arc  BD—9.A. 
Then  ^^= tan.  2^,  ^(7=  sec.  2^.  Draw 
CjE' bisecting  the  angle  J^  OS,  then  BE= 
tan. -4.  Also,  i?-£'=  tan.  ^,  because  the 
two  triangles  CBE  and  CDS,  are  in  all 
respects  equal. 

Now  by  the  similar  A's  ADE,  ABO, 
we  have  this  proportion, 

AD     \    DE    \    \    AB     ',    BO. 
That  is,    AD  :  tan.  A  :  :  tan.  2A  :  H 
Whence,     AD  •  i2=tan.  A  •  tan.  2 A 
By  adding  jK^  to  both  members  and  fac- 
toring, we  have 

(AD-{-E)E=  tan.  A  tan.  2^-|-i22 

But    (^i>+i^)=^(7=sec.  2^;  therefore, 
B  sec.  2^=tan.  A  tan.  2A-\-R^ . 


Q.  E.  D. 


(5.)  Show  that  in  any  plane  triangle,  the  base  is  to  the  sum  of  the 
other  two  sides,  as  the  sine  of  half  the  vertical  angle  is  to  the  cosine 
of  half  the  difference  of  the  angles  at  the  base. 

Let  ABC  be  the  A.  Call  AB  the 
base,  and  produce  A  C  the  shorter  side 
so  that  CD=:  OB  and  OEz=  OB.  Then 
if  0  be  taken  as  the  center  of  a  circle 
and  CB  radius,  that  circle  must  pass 
through  the  points  E,  B,  and  D,  and 
the  angle  EBD  must,  therefore,  be  a 
right  angle. 

Because  A  CB  is  the  exterior  angle 
of  the  A  ODB,  and  that  A  isosceles, 
the  angle  A  CB  must  equal  2D,  or  the 
angle  D  is  half  the  vertical  angle. 

Because  BA  0  is  the  exterior  angle  of  the  A  AEB,  we  have 
BAC=AEB+ABE  (1) 

But  AEB=CBE=CAB+ABE.  This  value  of  AEB,  sub- 
stituted in  ( 1 ),  gives  BA  0=  CBA+ABE+ABE    (2) 

Whence,  BAC^CBA=^2ABE  (3) 


TRIGONOMETRY.    ;  Ml 

This  last  equation  shows  us  that  ABE  is  half  the  difference  of 
the  angles  at  the  base. 

Now  in  the  A  ABD,  we  have 

AB    :     AD     :    :     sm.D     :     sin.  ^^i9. 

But  the  sin.  ABD=cos.  ABE^  because  the  sum  of  these  two 
angles  make  90°.     Hence  the  preceding  proportion  becomes 
AB  :  {AC-\-CB)  :  :  ^m.\ACB  :  gob.\{BAC--CBA).    Q.E.D. 

ScHO.  1 .     The  A  AEB  gives  us  this  proportion, 
AB     :     AE    :   :     sin.  E    :     sin.  ABE. 

Because  the  angles  E  and  D  together  make  90°,  sin.^=cos.i>. 

Hence,         AB     :     AE    :    :     cos.  D     :     sin.  ABE. 

That  is,  in  relation  to  the  triangle  ABC,  and  generally, 

The  base  of  any  'plane  triangle,  is  to  the  difference  of  the  other  two 
sides,  as  the  cosine  of  half  the  angle  opposite  to  the  base,  is  to  the  sine 
of  half  the  difference  of  the  other  two  angles.'^ 

ScHO.  2.  Draw  AH  parallel  to  EB,  and  of  course  perpendic- 
ular to  DB ;  then  we  have 

DA     '.    AE    \    '.     JDH    \    HB. 

If  Affhe  made  radius,  DR  is  tangent  to  the  angle  DAB,  and 
BB  is  tangent  to  the  angle  BAB. 

Because  ^^is  parallel  to  EB,  the  angle  BAB  is  equal  to  ABE; 
but  ABE  has  been  demonstrated  to  be  equal  to  the  half  difference 
of  the  angles  CAB,  CBA  ;  therefore,  DAB  is  the  half  sum  of 
the  same  angles,  for  the  half  sum  and  half  difference  of  any  two 
quantities  make  the  greater  of  the  two.  Therefore,  the  preceding 
proportion  becomes  the  following  theorem  : 

As  the  sum  of  the  sides  is  to  their  difference,  so  is  the  tangent  of  the 
half  sum  of  the  angles  at  the  base,  to  the  tangent  of  half  their  dif- 
ference. 

This  theorem  is  demonstrated  in  some  form  in  every  treatise  on 
plane  trigonometry.  It  is  the  7th  prop.,  page  149,  Robinson's 
Geometry 

(6.)  The  diference  of  two  sides  of  a  triangle,  is  to  the  difference  of 
the  segments  of  a  third  side,  made  by  a  perpendicular  from  the  oppo- 
site angle,  as  the  sine  of  half  the  vertical  angle  is  to  the  cosine  of  half 
the  difference  of  the  angles  at  the  base  ;  required  the  proof 

*This  is  theorem  v,  Robinson's  Geometry,  page  220.  , 


«66  ROBINSON'S  SEQUEL. 

Let  ABC  he  the 
A.  On  the  shor- 
ter side  CB  as  ra- 
dius, describe  a 
circle,  cutting  AB 
in  F,  AC  in  B, 
and  produce  AC 
to  K  Draw  CD 
perpendicular  to  the 
base,  then  DB  is  one  segment  of  the  base,  AD  is  the  other,  and 
AF  is  their  difference.  AJI  is  obviously  the  difference  of  the 
sides. 

Now  in  the  A  ABF,  we  have 

Aff    :    AF    :   :     sin.  AFff    :    sm.  AHF     (1) 

This  proportion  demonstrates  the  theorem,  as  will  appear  when 
we  show  the  values  of  these  angles. 

Because  CHFB  is  a  quadrilateral  in  a  circle,  the  angles 
HFB+E^im". 

But  BFB-\-AFir=lBO''. 

By  subtraction,     F—AFJI=0,  or  AFJI=F. 

In  the  same  manner  we  prove  that  AIIF=ABF. 

Substituting  these  equals  in  proportion  (1),  it  becomes 
AH    :    AF    :    :     sin.  ^^    :     sin.  ABF  (2) 

The  angle  F  is  half  the  angle  A  CB,  because  A  CB  is  at  the 
center  of  the  circle,  and  F  at  the  circumference  intercepting  the 
same  arc. 

Also,  sin.  ABF=cos.  ABIT,  because  EBE  is  a  right  angle,  and 
the  sine  of  an  arc  over  90°  is  equal  to  the  cosine  of  the  excess  over  90°. 

Again,  BCF—  the  sum  of  the  angles  at  the  base.     BHC,  or 
its  equal  CBH,  is  half  BCF\  therefore,  CBH  =  the  half  sum 
of  the  angles  at  the  base  of  the  triangle  A  CB,  and  HBA  is  their 
half  difference.     Whence  proportion  (2)  becomes 
Aff    :    AF    :  :    8m.^(ACB)     :     cos.:^( ABC—A).    Q.E.D. 

ScHO.    Because  -4  is  a  point  without  a  circle,  &c. 

ABxAF=AFxAff. 
Whence,        AB    :    AF    :  :    Aff    :    AF, 


TRIGONOMETRY.  «67 

The  two  A's  ABE,  AHF,  have  the  common  angle  A,  and  the 
sides  about  the  equal  angle  proportional,  therefore,  (th.  20,  b.  ii.) 
the  two  A's  are  similar,  and  AFH=E.    AHF=ABE. 

(7.)    Given  the  base,  the  difference  of  the  other  two  sides,  and  the 
difference  of  the  angles  at  the  base,  to  construct  the  triangle. 
(See  figure  to  Theorem  5.) 

Draw  AB  equal  to  the  given  base.  From  B  on  the  opposite 
side  of  the  base,  make  the  angle  ABE=z  half  the  difference  of 
the  angles  at  the  base. 

Take  AE,  the  given  difference  of  the  sides,  in  the  dividers  ; 
put  one  foot  on  A,  and  strike  an  arc  cutting  BE  in  E.  Join  AE, 
and  produce  EA. 

Make  the  angle  EBD=i90°.  BD  and  EA  produced  will  meet 
in  D.  Bisect  ED  in  C,  and  join  BG,  and  AOB  will  be  the  A 
required. 

N.  B.  This  problem  was  Suggested  bj  the  investigation  of 
theorem  v. 


(8.)  Prove  that  smr\l    ^    =tan.-\/-. 


Remark.  The  notation  sin.~*w,  signifies  an  arc  of  a  circle  whose 
radius  is  unity,  and  sine  u,  &c.,  &c. 

Hence  the  above  proposition  in  plain  English  is  this  : 

The  radius  of  a  circle  is  unity,  the  sine  of  an  arc  in  thai  circle  is 

^1 Prove  that  the  tangent  of  the  same  arc  must  be  - /- . 

^a-\-x  ^a 

Let  y=  the  cosine  of  the  arc  in  question. 

Then    ya+_^=l.     Whence,  y^^.JL-. 
a-\-x  a-\'X 

But  to  every  arc  we  have  the  following  proportion  : 
cos.     :     sin.     :    :     1     :    tan. 


That  is,   J-^    :    -/-^    :  :     1     :    tan. 
\a+x         ^a+x 

Or,        Ja    :     Jx    :   :     I     :    tan.=^-.    Q.  E.  D. 


268  ROBINSON'S  SEQUEL. 

(9.)  If    tan.(a-j)^,_sin^c^  ^^  tan.a  tan.4=  tan.^c. 
tan.  a  sin.^a 

To  perform  the  reduction,  multiply  by  tan.  a,  and  in  the  last 

term  take  its  equal  — '—  ;  then 
cos.a 

tan.  (a — 5)= tan.  a — _; '. . 

sin.  a  cos.a 

That  is,    ^1l^J?IL*_=tan.-        ^'"'"^ 


l-|-tan.atan.5  sin.a  cos.a 

tan.a-toii.J=tan.a+tan.=«  tan.4-C!in:!£+E5:!i^iL?i^) 

\  sin.a  cos.a  / 

Whence,         ( l+tau.=a)  tan.  j^sin.'c+sin.'otan^atoij 

sin.a  cos.a 
Multiplying  by  cos.a,  and  observing  that  (l-|-tan.^a)=sec.^a, 
and  cos.a  sec. a=l  ;  then 

,      ,     sin.^  c  .  sin.^ctan.atan.6 
sec. a  tan.o= + ' 


sin.a  sm.a 

Or,         sec.a  tan.6  sin.a=sin.^ c-|-sin.^  c  tan.a  tan.J. 

Take    — '—  for  tan.a  in  the  last  term,  then 
cos.a 

,       ,    .  .    „     ,  sin. ^c  sin.a  tan.  5 

sec.atan.6  sm.a=sm.^c4- 

cos.a 

Or,      (cos.a  sec.a)tan.5  sin.a=sin.2c  cos.a-j-sin.^c  sin.a  tan.6. 

Observing  again  that  (cos.a  sec.a)  =  l,  we  have 

tan.6  sin.a=sin.^c  cos.a-j-sin.^c  sin.a  tan.6 

Divide  each  term  by  cos.a,  and  taking  tan.a  for  ! — 1-,  we  find 

cos.a 

tan.a  tan.J=sin.^c-{-sin.^ctan.a  tan.^. 
(1 — sin.^c)tan.a  tan.5=sin.^c. 
But  (1 — sin.*c)=cos.2c,  because  sin.2c-|-cos.^c=l  ;  therefore, 
cos.  ^  c  tan.a  tan.5=sin.  ^  c 

tan.a  tan.6= '- — ^=tan.^c.     Q.  E.  D. 

cos.^c 


TRIGONOMETRY,  v  269 

SECTION   III. 

PROBLEMS  IN  SPHERICAL  TRIGONOMETRY  AND  ASTRONOMY 

Let  ABC  he  a  right  angled  triangle,  right 
angled  at  B.  a  the  side  opposite  A,  h  the 
side  opposite  B,  and   c  the  side  opposite  G. 

Taking  the  complement  of  the  oblique  an- 
gles A  and  C,  calling  them  A\  G\  and  the 
complement  of  h  calling  it  b\ 

Then  Napier's  Circular  Parts  give  us  the  following  equations. 
We  retain  the  same  numbers  for  the  equations  as  in  our  Geometry, 
page  186. 

(11)  i2  sin.c=tan.a  tan.  ^'  (16)  i?  sin.^'=tan.2>' tan.c 

(12)  i2sin.rt=tan.ctan.(7'  (17)  i2sin.^'=cos.a  cos.C" 

(13)  i2sin.a=cos.5'cos.^'  (1^)  i2sin.5'=cos.a  cos.c 

(14)  jRsimc=cos.5'cos.(7'  (19)  i2  sin.  C"=tan.5' tan.  a 

(15)  i2sin.5'=tan.^'tan.(7'  (20)  J^  sin.  (7'=cos.c  cos.^' 

These  equations  are  written  in  the  present  form  to  assist  the 
memory,  the  second  members  being  the  products  of  two  cosines 
or  two  tangents  ;  but  in  practice,  we  often  modify  an  equation  by 
taking  sine  for  cosine,  and  cotangent  for  tangent,  and  the  reverse. 

For  instance,  in  equation  (18),  we  invariably  take  cos.S  for 
sin.i',  it  being  the  same,  which  saves  the  trouble  of  finding  the  com- 
plement to  the  hypotenuse.  The  same  may  be  said  of  other  com- 
plements. 

In  all  spherical  triangles,  right  angled  or  oblique  angled,  the  sine  of 
the  sides  are  to  each  other  as  the  sines  of  the  angles  opposite  to  them. 

When  two  sides  of  a  spherical  triangle  are  given,  there  can  be 
but  one  result,  that  is,  there  can  be  no  ambiguity  about  the  parts 
required  ;  but  when  only  one  side  is  given,  and  one  of  the  ob- 
lique angles  in  a  spherical  triangle,  the  conditions  correspond 
equally  to  two  triangles,  and  the  answer  is  said  to  be  ambiguous. 
For  a  learner  fully  to  comprehend  this,  it  is  necessary  to  learn  to 
construct  his  triangles  as  follows  : 

We  shall  illustrate  by  examples,  beginning  with  the  10th  ex- 


270  ROBINSON'S  SEQUEL. 

ample,  page   199,  Robinson's  Geometry,  which  will  sufficiently 
illustrate  several  others. 

(1.)  In  the  right  angled  triangle  ABC,  right  angle  at  B,  given 
AB  29°  12'  50"  and  the  angle  C  37°  26'  21",  to  find  the  other  parts. 

To  construct  a  spherical  A, 
the  operator  should  have  a 
scale  of  chords  and  semitan- 
gents  ;  but  he  can  do  all  with 
a  ruler  and  dividers. 

Take  0(7  in  the  dividers 
equal  to  the  chord  of  60°,  (or 
any  distance  if  no  scale  is  at 
hand),  and  from  any  point  0 
as  a  center  describe  the  circle 
CHDh.  Draw  CD  and  Hh 
at  right  angles  through  the 
center.  Each  of  the  lines 
0  (7,  ODy  OH,  as  well  as  the  curve  HC,  HD,  &c.  represent  90° 
on  a  sphere.  OHC  is  a  right  angle, — that  is,  any  line  from  0  to 
the  circumference  will  make  a  right  angle  with  the  circumference- 

Now  from  C  we  propose  to  make  the  angle  JICA=3'7°  26'  21". 
Divide  the  quadrant  BJ)  into  degrees,  beginning  at  II.  Take 
IFF  equal  to  37°  26'  21" ;  or  if  the  scale  is  used,  take  the  chord 
of  37°  26'  21"  from  the  scale  and  apply  it  from  II  to  F.  Apply 
the  ruler  from  C  to  P,  and  through  the  point  n,  where  this  line 
crosses  ITO,  describe  the  curve  CnD. 

From  If,  set  off  IfQ=^29°  12'  50",  apply  the  ruler  between  Q 
and  C,  and  mark  F  where  this  line  cuts  HO.  From  the  center 
0,  with  0  V  radius,  strike  the  arc  VA.  Lastly,  through  A  and 
0  draw  BAG  through  the  center.  The  A  ABC  or  its  supple- 
ment DBA,  is  the  one  required.  The  side  A  C  is  measured  by 
the  arc,  but  neither  A  C  nor  the  angle  A  can  be  measured  instru- 
mentally.  To  measure  sides,  they  must  either  be  on  the  circum- 
ference or  on  the  straight  lines  through  the  center. 

Remark.  If  the  angle  BAC  had  been  given,  we  should  call 
the  triangle  ADQ  the  supplemental  triangle,  for  ^6^  is  the  sup- 


TRIGONOMETRY.  271 

plement  to  AB,  AD  to  AC,  and  the  angle  ABO  is  supplemental 
to  ADB  or  its  equal  BCA. 

When  we  have  all  the  parts  of  the  triangle  ABC,  we  in  effect 
have  all  the  parts  of  the  triangle  DAB,  also  all  parts  of  the  A 
AD  G  and  all  parts  of  the  triangle  GA  C. 

That  is,  when  one  spherical  triangle  is  determined,  we  have 
three  others,  the  whole  four  making  up  a  hemisphere. 

For  the  numerical  computation  oi  AC  we  take  equation  (14) 

modified  thus : 

cir.   jn    c'.r.  7.     -Ssin.c 19.688483 

sm.  A  C=sm.  6= — -, — — 

sm.(7 9.783843 

sin.  63°  24'  13" 9.904640 

To  find  BC  or  a,  we  take  a  modification  of  (18). 

E  cos.5 19.775374 

cos.a= 

cos.c   9.940917 

cos.  46°  56'  2" 9.834457 

To  find  the  angle  A,  we  take  a  modification  of  equation  (13). 

.      .     Bsin.a 19.863539 

sm.-4= — — — 

sm.b   9.904640 

sin.  C6°  27'  60" • 9.958899 

Whence, 
^C=:53°  24'  13",  BC=46°  55'    2",  and  the  Z-  A,  65°    27'  60". 
AD=126°  36'  47",  ^2>=133°  4'  58",  and  theA  BAD,  114°  32'  10" 

The  same  figure  will  sufficiently  illustrate  example  12,  page 
199,  Robinson's  Geometry. 

(2.)  In  the  right  angled  triangle  ABC,  given  AB,  64°  21'  36", 
and  the  angle  C,  61°  2'  16",  to  find  the  other  parts. 

(14)        sin.^C7=sin.6=^^^ l^M^nS 

^     '  sm.  C 9.941976 

sin.  68°  16'  16" 9.967940 

Whence,    -4(7  ==  68°  15'  16",  and  ^i>  =  111°  44'  44".      The 

answers  given  in  the  book  correspond  to  the  triangle  ADB, — • 

and  those  answers  were  given  to  exercise  the  judgment  of  the 

learner. 

The  other  parts  are  found  as  in  the  last  example. 


272  ROBINSON'S  SEQUEL. 

(3.)  In  the  right  amjled  spherical  triangle,  given  AB,  100®  10'  3" 
and  the  angle  BCA,  90°  14'  20",  to  find  the  other  parts. 

Because  the  sines,  cosines, 
(kc,  of  the  tables  correspond 
to  arcs  under  90°  ;  therefore 
we  will  operate  on  the  sup- 
plemental triangle,  ADE. 
BC^DE.  180°— .^j5=79° 
49'  67"= JZ>=c. 

The  angle  ^7)^=90°— 
(14'20")=89°45'40". 

AC=h,  and  AB  =  h'  in 
the  equations.  AB=^Cy  AED 
=90°,  ^i>^=:(7'  =  89°46' 
40". 

To  solve  this  A,  we  use  equation  (20). 

R  cos.  C 17.620026 


sin.  A 


COS.  c   9.246810 

sin.  1°  2ri2" 8.373216 


To  compute  AI>y  we  take  equation  (16)  ;  AB  the  supplement 
of  AC=h. 

R  COS.  5=cot.  A  cot.  C. 

cot.^=l°21'  12" 11.626819 

cot.(7=89°46'40" 7.619860 

AB  COS.  79°  60'    6" 9.246679 

AC        100°    9' 65" 

To  find  BE,  or  its  equal  BC,  we  take  equation  (13). 

i2sin.a=  sin.6sin.-4. 

sin.  79°  60'    6" 9.993128  ' 

sin.    1°  21'  12" 8.373216 

BC,  sin.  1°  19' 62" 8.366344 

These  examples  give  a  sufficient  key  to  the  solution  of  all  other 
examples  in  right  angled  spherical  trigonometry.  * 


TRIGONOMETRY.  273 

We  now  turn  to  the  application  of  spherical  trigonometry — 
taking  the  examples  from  page  215,  Geometry. 

MISCELLANEOUS  ASTRONOMICAL  PROBLEMS. 

(1.)  In  latitude  40°  48'  north,  the  sun  hore  south  78°  16'  west,  at 
3h.  38m.  P.  M.,  apparent  time.  Required  his  altitude  and  declina- 
tion, making  no  allowance  for  refraction. 

Ans.  The  altitude,  36°  46',  and  declination,  15°  32'  north. 

Let  Hh  be  the  horizon, 
Z  the  zenith  of  the  observer, 
P  the  north  pole,  and  PS 
a  meridian  through  the  sun. 

PZ  is  the  co-latitude,  49° 
\9.\  and  PS  is  the  co-decli- 
nation or  polar  distance,  one 
of  the  arcs  sought.  ZS  is 
the  co-altitude  or  ST  is  the 
altitude  of  the  sun  at  the 
time  of  observation. 

The  angle  ZPS  is  found 
by  reducing  3h.  38m.  into 
degrees  at  the  rate  of  4m.  to  one  degree  ;  hence,  ZPS=54°  30' 

Because  EZS=7Q°  16',  PZS=101°  44'.  From  Z  let  fall  the 
perpendicular  Z^  on  PS.  Then  in  the  right  angled  spherical 
A  PZQ,  equation  (13)  gives  us* 

P  sin.  Z^=sin.  PZsin.  P. 

sin.  PZ=sin.  49°  12' .9.879093 

sin.  P  =sin.  54°  30' 9.910686 

sin.  Z^=sin.  38°  2'  33" 9.789779 

To  obtain  the  angle  PZQ,  we  apply  equation  (19),  which  gives 

P  COS.  PZQ=cot.  PZ  tan.  ZQ, 
That  is,  i2cos.  PZQ=t3,n.  40°  48'  ta,n.  38°  2'  33". 

*  To  apply  the  equations  ■witliout  confusion,  letter  each  right  angled  spher- 
ical triangle  ABC,  right  angled  at  B,  then  A  must  be  written  in  place  of  P ; 
nndwhen  operating  on  ZSQ,  write  A  in  place  of  S,  and  C  for  the  angle  SZQ 
18 


io 


^4  .       ROBINSON'S  SEQUEL. 

9.936100 
9.893464 

PZQ=  COS.  47°  30'  50' 9.829564 

PZS=        101°  44' 
SZQ=         54°  14'  10" 
To  obtain  ZS  or  its  complement,  we  again  apply  (19) 

(19)         E  COS.  SZQ=cot.  ZStsLYL.  ZQ. 
That  is,  i2cos.  54°  14'  10"=tan.  STtan.38°  2'  33". 

i?  COS.  54°  14'  10"= 19.766744 

tan.  38°  2'  33 "      = 9.893464 

tan.  36°  46',  nearly, 9.873280 

To  find  PS,  we  take  the  following  proportioif  : 

sin.  P     :     sin.Z^     :    :     sm.  PZS     :     &m.  PS 
That  is,  sin.  54°  30'  :  cos.  36°  46'  :  :  sin.  101°  44'  :  sin.  PS 

COS.  11°  44' 9.990829 

cos.  36°  46' .9.903676 

19.894505 

sin!  54°  30' 9.910686 

PS,  74°  28'  sin 9.983819 

Whence,  the  sun's  distance  from  the  equator  must  have  been 
15°  32' north. 

(2.)  In  north  latitude,  when  the  sun's  declinatlo7i  ?m5  14°  9.Q' north, 
his  altitudes,  at  two  different  times  on  the  same  forenoon,  were  43° 
7'-}-,  (ind  67°  10'-(-  ;  and  the  change  of  his  azimuth,  in  the  inter' 
vol,  45°  2'.     Required  the  latitude.  Ans.  34°  20'  north. 

Let  PK  be  the  earth's   axis,     ^^^^^^^ 
Qq  the  equator,  and  Bh  the  ho-    ^KUBm^^ 

Also,  let  Z  be  the  zenith  of  the  ^■^•^^^ 
observer,  Sm  the  first  altitude, 
Tn  the  second,  and  the  angle 
rZ>S=45°  2'.  Our  first  opera- 
tion must  be  on  the  triangle  ZTS. 
ZT=22°  50',  Z^=46°  53',  and 
we  must  find  TS,  and  the  Z_  TSZ- 


TRIGONOMETRY.  275 

From  T,  conceive  TB  let  fall  on  ZS  making  two  right  angled 
A's  ;  and  to  avoid  confusion  in  the  figure,  we  will  keep  the  arc 
TB  in  mind,  anA  not  actually  draw  it. 

Then  the  A  ZTB  furnishes  this  proportion  : 

R     :     sin.  22°  60'     :    :     sin.  46°  2'     :     sin.  TB=sm.  16°  66'  8" 

To  find  ZB  we  have  the  following  proportion,  (see  p.  186  Geo.) 
R     :     COS.  ZB     :    :     cos.  16°  66'  8"     :     cos.  22°  60' 

Whence,  we  find  Z5=16°  34'  20".  Now  in  the  right  angled 
spherical  A  TBS,  we  have  TB  =  15°  56'  8",  BS=46°  63'— 16° 
34'  20",  or  ^5=30°  18' 40" ;  and  TS  is  found  from  the  following 
proportion : 

R     :     COS.  15°  56'  8"     :    :     cos.  30°  18'  40"     :     cos.  TS 

This  gives  TS=33°  53'  16".  To  find  the  angle  TSZ,  we  have 
the  proportion,  sin.  33°  53' 1 6"  :  R  :  :  s'm.  TB  15°  66' 8"  :  sin.  TSZ. 

Whence,  the  angle  TSZ=29''  30'. 

The  next  step  is  to  operate  on  the  isosceles  spherical  A  PTS, 
We  require  the  angle  TSF. 

Conceive  a  meridian  drawn  bisecting  the  angle  at  P,  it  will 
also  bisect  the  base  TS,  forming  two  equal  right  angled  spherical 
triangles. 

Observe  that  P>S^=75°  40'  and  ^  TS=16°  56'  38". 

To  find  the  angle  TSF  we  apply  equation  (19),  in  which  a= 
16°  56'  38",  5=75°  40',  and  the  equation  becomes 

R  cos.  TSP—coi.  76°  40'  tan.  16°  56'  38" 

Whence,  TSP=S5''  31'  40",  and  PSZ=Q5°  31'  40"— 29°  30' 
=66°  1'  40". 

The  third  step  is  to  operate  on  the  A  ZSP ;  we  now  have  its 
two  sides  ZS  and  SP,  and  the  included  angle. 

From  Z  conceive  a  perpendicular  arc  let  fall  on  SP,  calling  it 
ZB  ;  then  the  right  angled  spherical  triangle  SZB,  gives 

R     :     sin.  ZS     :    :     sin.  Z SB     :     sin.  ZB 
That  is,  R  :  sin.  46°  53'  :  :  sin.66°  1'40"  :  sin.Z^=sin.37°  15'20" 
To  find  SB  we  have  the  following  proportion,  (see  Geo.  p.  185.) 

R     :     cos.  SB     :    :     cos.  ZB     :     cos.  ZS 
Thai  is,  R  :  cos.  SB  :  :  cos.  37°  15'  20"  :  cos.  46°  63' 


276 


ROBINSON'S   SEQUEL. 


Whence  SB=30''  49'  40".  Now  from  FS,  15"  40',  take  SB, 
30°  49'  40",  and  the  diflference  must  be  BP,  44°  50'  20". 

Lastly,  to  obtain  PZ,  and  consequently  Z  Q  the  latitude,  we  have 
R     :     COS.  ZB     :     cos.  BP     :     cos.  ZP==sin.  ZQ 

That  is,  B  :  cos.  37°  16'  20"  :  :  cos.  44°  50'  20"  :  sin.  ZQ= 
sin.  34°  21' north. 

This  computation  differs  one  mile  from  the  given  answer,  but 
any  two  operators  will  differ  about  this  much,  unless  each  observe 
the  utmost  nicety. 

This  is  a  modification  of  latitude  by  double  altitudes,  but  in 
real  double  altitudes  the  arc  ^aS^  is  measured  from  the  elapsed 
time  between  the  observations,  and  the  angle  TZS  is  not  given. 


(3.)  In  latitude  16°  4'  north,  when  the  su7i's  declination  is  23°  2' 
north.  Mequired  the  time  in  the  afternoon,  and  the  sun's  altitude  and 
hearing  when  his  azimuth  neither  increases  nor  decreases. 

Ans.  Time,  3h.  9m.  26s.  P.  M.,  altitude,  45°  1',  and  bearing 
north  73°  16'  west. 

Let  Pp  be  the  earth's  axis, 
Hh  the  horizon,  Qq  the  equator, 
QZ  and  Pp,  each  equal  to  16° 
4'  north,  and  Qd,  qd,  each  e- 
qual  to  23°  2' ;  then  the  dotted 
curve  dd  represents  the  parallel 
of  the  sun's  declination. 

Through  Z  and  N  an  infinite 
number  of  vertical  circles  can 
be  drawn,  one  of  these  will  touch 
the  curve  dd  ;  let  it  l^e  Z  OK 

At  the  point  0  where  this  circle  touches  the  curve  dd  will  be 
the  position  of  the  sun  at  the  time  required,  and  P  OZ  will  be  a 
right  angled  spherical  A,  right  angled  at  0.  The  problem  re- 
quires the  complement  of  ZO,  and  the  time  corresponding  to  the 
angle  ZPO. 

In  the  spherical  A  P  OZ,  we  have 

R     :     COS. PO     :    :     cos.  ZO     :     cos.  PZ 

That  is,  R     :     sin.  23°  2'     :    :     sin.  altitude     :     sin.  16°  4' 


Whence,  sin.  alt.  = 


TRIGONOMETRY, 
i?  sin.  16°  4' 


277 


sin.  46°  r  nearly'.  Ans. 


sin,  23°  2' 
To  find  the  angle  at  P,  we  have  the  following  proportion  : 

COS.  16°  4'     :     R     :    :     cos.  46°  1'     :     sin.  P 

Whence,  sin.  P  =  sin.  47°  21' 30",  and  ZPO  =  47°  21' 30", 
which  being  changed  into  time,  at  the  rate  of  16°  to  one  hour, 
gives  3h.  9m.  26s. 

To  find  the  angle  PZ  0,  we  have  this  proportion  : 
cos.  16°  4'     :     R     '.    :     cos.  23°  2'     :     sin.  PZO  =  sin.  73°  16' 

(4.)  The  sunset  south-west  ^  sovthy  when  his  declination  was  16° 
4'  south.     Required  the  latitude.  Ans.  69°  1'  north. 

Draw  a  circle  as  before.    Let 
Hh  be  the  horizon,  Z  the  zenith,     ^^^vn 
P  the  pole.      The  great  circle 
PZH'i^  the  meridian,  and  ZCN 
at  right    angles  to  it,    and   of 

coui'se  east  and  west.  Let  BC  ^^^S^^ff^BBi^^KKKUi 
be  a  portion  of  the  equator,  and  ^^H^H|HBn^S^^HI 
B  0  the  arc  of  declination.  The  ^^|SHH^H|^H^SBI 
position  on  the  horizon  where  ^^^|^8^^^H^|B^B^| 
the  sun  set  is  the  arc  110=45°  ^^^B^l^^^P^^Bs^^M 
—6°  37'  30"=39°  22'  30". 

Consequently,  the  arc  00=50°  37'  30".      , 

In  the  right  angled  spherical  triangle  BOO,  we  have  BC,  BO 
given  to  find  the  angle  BOO,  which  is  the   complement  of  the 
latitude,  or  the  complemc'nt  of  the  angle  B  CZ. 
To  find  the  angle  BOO,  we  apply  equation  (14). 

i?  sin.  ^  0=sin.  OCsin.  J5(70 
That  is,  R  sin.  16°  4'=sin.  50°  37'  30"  sin.  BOO 

Rsin.  16°  4' 19.442096 

sin.  50°  37'  30" 9.888184 

cos.  69°  1'  nearly, 9.653912 

ScHO.  The  arc -6  (7  on  the  equator  measures  the  angle -BP (7, 
corresponding  to  the  time  from  6  o'clock  to  sun  rise  or  sun  set. 


S78  ROBINSON'S  SEQUEL. 

This  arc  is  called  the  arc  of  ascensional  difTerence  in  astronomy. 
The  time  of  sun  set  is  before  six  if  the  latitude  is  north  and  tlie 
declination  south,  as  in  this  example,  but  after  six,  if  the  latitude 
and  declination  are  both  north  or  both  south. 

To  obtain  this  arc,  the  latitude  and  declination  must  be  given  ; 
that  is,  BO  and  the  angle  BCO,  the  complement  of  the  latitude. 
Here  we  apply  (12),  that  is, 

M  sin.  BO  =  tan.  D  tan.  L 
an  equation  in  which  D  represents  the  declination,  and  L  the 
latitude. 

(5.)  The  altitude  of  the  suUy  when  on  the  equator,  was  14°  28'-]-, 
hearing  east  22°  30'  south.     Required  the  latitude  and  time. 

Ans.  Latitude  56°  T,  and  time  7h.  46m.  12s.  A.  M. 

Let  S  be  the  position  of  the  sun  on  the  equator.  (See  the  last 
figure.)  Draw  the  arc  ZS,  and  the  right  angled  spherical  A 
ZQS  is  the  one  we  have  to  operate  upon. 

Then  ZS  is  the  complement  of  the  given  altitude,  and  the  an- 
gle QZS,  is  the  complement  of  22°  30'.  The  portion  of  the 
equator  between  Q  and  S,  changed  into  time,  will  be  the  required 
time  from  noon,  and  the  arc  QZ  will  be  the  required  latitude. 

First  for  the  arc  QS. 

R     :     sin.  ZS    :    :     sin.  QZS    :     sin.  QS 
That  is,  R  :  cos.  14°  28'  :  :  cos.  22°  30'  :  sin.  ^aS^  =  73°  27'38" 

But  73°  27'  38"  at  the  rate  of  4m.  to  one  degree,  corresponds 
to  4h.  13m.  48s.  from  noon, — and  as  the  altitude  was  marked  -|-, 
rising,  it  was  before  noon,  or  at  7h.  46m.  12s.  in  the  morning. 

To  find  the  arc  QZ  we  have  the  following  proportion  : 

R     :     COS.  63°  27' 38"     :    :     cos.  ^Z     :     sin.  14°  28' 

Whence,  cos.  ^Z=cos.  66°  1'  nearly,  and  56°  1'  is  the  latitude 
souffht. 

o 

(6.)  The  altitude  oft/ie  sun  was  20°  41'  at  2h.  20m.  P.  M.  when 
his  declination  was  10°  28'  south.  Required  his  azimuth  and  the 
latitude.     Ans.  Azimuth  south  37°  5'  west,  latitude  51°  58'  north. 


TRIGONOMETRY.  279 

This  problem    furnishes   the     

spherical  A  PZ  0,  in  which  the     ^H!|^^B9I 

side  Z  0  is  the  complement  of    ^B|fig8Q^&^^|B| 

20""  41'  or  eO""  W,  F0=90''  ^H^j^KKBrn 
-\-iO°  28'  =  100°  28',  and  the  ||yi^|^SIHBI 
angle  ZFO  is  2h.  20m.,  chang- 
ed into  degrees  at  the  rate  of 
15°  to  one  hour,  or  ZPO=35°. 
Now  in  the  triangle  ZFO, 
we  have 

sin.  ZO     :     sin.  ZPO     :    :     sin.PO     :     sin.PZO      That  is, 
cos.  20°  41'  :  sin.  35°  :  :  cos.  10°  28'  :   sin.  BZO  =  cos.  37°  5'. 
In  the  right  angled  spherical  A  B  OZ,  we  apply  equation  (16). 
(16).         R  cos.37°  5'=tan.  20°  41'  tan.  BZ. 

i2cos.  37°  6' 19.901872 

tan.  20°  41' 9.576958 

tan.  ^Z=tan.  64°  40'  40" 10.324914 

To  find  FB  in  the  right  angled  A  BFO,  we  apply  the  same 
equation,  (16).     R  cos.  35°=tan.  10°  28'  tan.  FB. 

R  cos.  35° 19.913365 

tan.  10°  28' 9.266555 

tan.  12°  42'  40" ..10.646810 

But  FB  is  obviously  greater  than  90°,  therefore  the  point  B  is 
12°  42'  40"  below  the  equator,  but  from  jB  to  Z  is  64°  40'  40"; 
therefore  from  Z  to  the  equator,  or  the  latitude,  is  the  difference 
between  64°  40'  40"  and  12°  42'  40",  or  51°  58'  north. 

Ans.  Lat.  51°  58'  north. 

(7.)  If  in  August  1840,  Spica  was  observed  to  set  2h.  26m.  14s. 
hefore  Arcturus,  what  was  the  latitude  of  the  observer  ?  Taking  no 
account  of  the  height  of  the  eye  above  the  sea,  nor  of  the  effect  of 
refraction.  Ans.   36°  48'  north. 

By  a  catalogue  of  the  stars  to  be  found  in  the  author's  Astron- 
omy, or  in  any  copy  of  the  English  Nautical  Almanac,  we  find  the 
positions  of  these  stars  in  1 840  to  have  been  as  follows  :  ^ 

Spica,  right  ascension,  13h.  16m.  46s.     Dec.  10°  19' 40"  south. 

Arcturus,  "         *'  14h.    8m.  25s.     Dec.  20°    1'    4"  north. 


280  ROBINSON'S  SEQUEL. 

Let  L  =  the  latitude  sought.  Put  d=10°  19'  40",  and  D— 
20°  r  4". 

The  difference  in  right  ascensions  is  61m.  39s.,  and  this  would 
be  about  the  time  that  Arcturus  would  set  after  Spica,  provided 
the  observer  was  near  the  equator  or  a  little  south  of  it ;  but  as 
the  interval  observed  was  2h.  26m.  14s.,  the  observer  must  have 
been  a  considerable  distance  in  north  latitude.  In  high  southern 
latitudes  Arcturus  sets  before  Spica. 

When  an  observer  is  north  of  the  equator,  and  the  sun  or  star 
south  of  it,  the  sun  or  star  will  set  within  six  hours  after  it  comes 
to  the  meridian. 

When  the  observer  and  the  object  are  both  north  of  the  equa- 
tor, the  interval  from  the  meridian  to  the  horizon  is  greater  than 
six  hours. 

The  difference  between  this  interval  and  six  hours,  is  called  the 
ascensional  difference,  and  it  is  measured  in  arc  hj  £0  in  the 
figure  to  the  4th  example. 

Now  let  X  =  the  ascensional  difference  of  Spica  corresponding 
to  the  latitude  £,  and  y  =  the  ascensional  difference  correspond- 
ing to  the  same  latitude  ;  then  by  the  scholium  to  the  4th  exam- 
ple, calhng  radius  unity,  we  shall  have 

sin.  a:=tan.  L  tan.  d  ( 1 ) 

sin.  y^tan.  L  tan.  J)  (2) 

The  star  Spica  came  to  the  observer's  meridian  at  a  certain  time 
that  we  may  denote  by  M. 

Then  Jlf-^/^e— —  )  =  the  time  Spica  set. 

And  M-{-51m.  39s.4-(6-|-^  )=  the  time  Arcturus  set. 

By  subtracting  the  time  Spica  set  from  the  time  Arcturus  set 
we  shall  obtain  an  expression  equal  to  2h.  26m.  14s.     That  is 

51m.  39s.  +^+i^=2h.  26m.  14s. 

Ot,  -^+I_=lh.  34m.  36s.  (3) 

'  16^16  ^  ^ 

^  ar+y=16(lh.  34m.  36s.)  (4) 

Equation  (3)  expresses  time.     Equation  (4)  expresses  arc. 

When  we  divide  arc  by  16  we  obtain  time,  one  degree  being 


TRIGONOMETRY.  281 

the  unit  for  arc,  and  one  hour  the  unit  for  time  ;  therefore,  when 
we  multiply  time  by  15  we  obtain  arc ;  that  is,  Ih.  multiplied  by 
15  gives  15°  ;  hence  (4)  becomes 

a;4-y=23°  39'=a 
x—a—y  (5) 

That  is,  the  arc  x  is  equal  to  the  difference  of  the  arcs  a  and  y ; 
but  to  make  use  of  these  arcs  and  avail  ourselves  of  equations 
(1)  and  (2),  we  must  take  the  sines  oi  the  arcs,  (see  equation 
(8),  plane  trigonometry)  ;  then  (5)  becomes 

sin.  a:=sin.  a  cos.  y — cos.  a  sin.  y  (6) 

Substituting  the  values  of  sin.  x  and  sin.  y  from  (1)  and  (2), 
(6)  becomes 

tan.  L  tan.  c?=sin.a  cos.  y — cos.  a  tan.  L  tan.  D    (7) 
Squaring  (2),         sin.2y=tan.^i/tan.2i). 
Subtracting  each  member  from  unity,  and  observing  that  (1 — 
sin.^y)  equals  cos.^y,  then 

cos.^2/=l — tan.  ^Z  tan.  2  i). 
Or,  cos.  y=  ^1— tan.^i/tan.^i). 

This  value  of  cos.y  put  in  (7),  gives 
tan.  Ztan.  <^=sin.  aj\ — tan.^Z  tan.^i> — cos.  «tan.  Xtan.  D    (8) 
By  transposition  and  division, 

/tan.rf+cos.atan^\  tan.  i;=Vl-tan.=itan.=i) 
\  sin.  a  / 

Squaring,   (^g^^lJ+g"^- "*'"'• -"Vtan.^X==l-tan.'X  tan.'i) 
\  sin.  a  J 

Dividing  by  tan.^X  and  observing  that    _-  =  cot.^Z   we 

lan.  j-i 

have  /tan^+cos^tan^\  =oot.=i-tan.^/) 

\  sin.  a  / 

Or,  cot.'X=tan.'2)+('-ggl^''"^-  "^.g^^-gV 

\  sin.  a  / 

=tan.^2)+C^+'^-V 
Vsin.  a      tan.  a  / 

We  must  now  find  the  numerical  value  of  the  second  member. 
Using  logarithmic  sines,  cosines,  tangents,  (fee,  we  must  diminish 
the  indices  by  10,  because  the  equation  refers  to  radius  unity, 
log.  tan.  Z>.==— 1.561460.  tan.-i>=— 1.122920=0.132712  num. 


1282  V      ROBINSON'S  SEQUEL. 

log.  tan.  d —1.260623    log.  tan.  i> —1.561460 

sin.  a — 1.603305        tan.  a — 1.641404 


0.45424 —1.657318    0.83188. . . .  . .  —1.920056 

0.45424-1-0.83188=1.28612         (1.28612)2  =  1.654105 
Whence,  cot.2Z=0.132712-|^l. 654105=  1.786817 

Square  root,      ..cot.  Z=  1.33672 

Taking  the  log.  of  this  number,  increasing  its  index  by  10  will 
give  the  log.  cot.  in  our  tables. 

log.  1.33672=0.126076-f-10.=10.126076=cot.  36°  48' 

(8.)  On  the  14th  of  November,  1829,  Merikar  was  observed  to  rise 
48m.  3s.  before  Aldebaran  :  what  was  the  latitude  of  the  observer  ? 

Ans.  39°  34'  north. 

The  positions  of  these  two  stars  in  the  heavens,  Nov.  1829, 
were  as  follows : 

Menkar,  right  ascension,  2h.  53m.  21s.      Dec.    3°  24'  52"  north. 
Aldebaran,  "  4h.  26m.    7s.      Dec.  16°  19' 31"  north. 

Aldebaran  passes  the  meridian  Ih.  32m.  46s.  after  Menkar. 
Now  let  M  represent  the  time  Menkar  was  on  the  meridian,  then 
M-\-\\i.  32m.  46s.  represents  the  time  Aldebaran  was  on  the 
meridian.  Also,  let  x=  the  arc  of  ascensional  diflference  corres- 
ponding to  the  latitude  and  the  star  Menkar,  and  y  that  of  the 
star  Aldebaran. 

Then  M—(q-\-—\  —  the  time  Menkar 

And    Jf-l-lh.  32m.  46s — (  6-[-  -  )  =  the  time  Aldebaran  rose. 

Subtracting  the  upper  from  the  lower,  the  difference  must  be 
«8m.  3s.  ;  that  is, 

lh.-|-32m.  46s 'L-X-—=A^m,  3s. 

^  16  '  15 

Whence,  -^— !_=  —44m.  43s.=  —0.74527. 

16     16 

That  is,  Ih.  being  the  unit,  44m.  43s.  =  0.74527  of  an  hour, 

and  multiplying  by  1 5,  we  shall  have  as  many  degrees  of  arc  as 

we  have  units  ;  therefore, 

a;— y=— (0.74527)15=— 11°  10'  45"=— a. 

x=y—a. 

sin.  a:=sin.^  cos.o — cos.y  sin.a  ( 1 ) 


rose. 


TRIGONOMETRY.  283 

Put  c/=3°  24'  52",  D=\6°  19'  31",  and  L=  the  required  lat- 
itude.    Then  by  scholium  to  the  4th  example, 

sin.  a;=tan.c?  tan.Z.         sin.y=tan.i>  tan.Z. 
These  values  of  sin.a:  and  sin.y,  substituted  in  (1),  give 
tsm.d  tan.Z=cos.a  tan.D  tan.Z — cos.y  sin.a         (  2) 
But  sin.2y==tan.2i>tan.2i;,  and  1— -sin.2y=l— tan.^i^tan.^Z. 
Or,  cos.^y=l — ^tan.^*Z)tan.2j&. 

Or,  cos.y=iJl — ^tan.^2>tan.^X. 

By  substituting  this  value  of  cos.y  in  (2)  and  transposing,  we 

find 

sin.a^l — tan.^i)  tan.2j&=(cos.a  tan.i) — tan.e?)tan.X 

Dividing  by  sin.a,  and  observing  that  -r-^= ,  we  have 

"^    •'  sm.  a    tan.a 

n — 1 — rm — rr     /tan.i)    tan.c?\  ,       r 

J\ — tan.^i)  tan.2Z=l — )  tan.ii. 

\  tan.a      sm.a/ 

Squaring  and  dividing  by  tan.^Z,  and  at  the  same  time  observ- 

insr  that =cot.Z,  and  we  shall  have 

^  tan.Z 

cot.«Z-tan.«i>=f-^^-*_^V 
\  tan.a      sin.  a/ 

We  will  now  find  the  numerical  values  of  the  known  quantities. 

Log.  tan.i). .  .—1.466696        Log,  tan.c?. . . — 2.776685 

Log.  tan.a —1 .296 1 79         Log.  sin.a 1 .2876 1 7 

Log.  1 .48089 .  . .     0.170517     Log.  0.3076 . . .  —1.488068 

tan.2i)=0.085778  1.48089—0.3076=1.17329 

Whence,  cot.^  Z— 0.085778= (1.1 7329)  2. 

Or,  cot.2ii=   1.462293. 

cot.Z=   1.20925. 
Log.  cot.i;+10i=10.082785=cot.  39^  34'.     Ans, 

(9.)  iw  latitude  16°  40'  north,  when  the  sun's  declination  was  23° 
1 8'  northy  I  observed  him  twice,  in  the  same  forenoon,  bearing  north 
68°  30'  east.  Required  the  times  of  observation,  and  his  altitude  at 
each  time. 

Ans.  Times  6h.  15m.  40s.  A.  M.,  and  lOh.  32m.  48s.  A.  M., 
altitudes   9°  69'  36",  and  68°  29'  42". 


2B4  ROBINSON'S  SEQUEL. 

Let  Z  be  the  zenith,  P  the 
north  pole,  and  the  curve  dd  be 
the  parallel  of  the  sun's  declina- 
tion along  which  it  appears  to 
revolve.      Make  the  angle  PTiS' 

equal  to  68°  30' ;  then  the  sun  bii^^^^^^^^^^I 
was  at  S  at  the  time  of  the  first  HV^^^BHHH^^^H 
observation,  and  at  S'  at  the  time     IIS^^^^HH|B|HI 

the  ^I^S^^BBB^BSfl 

In  the  spherical  A  PZS'  there     ^^^I^^H^bIB^^I 
is  given  PZ,  PS'  and  the  angle 
PZaS"  ;  also,  in  the  A  PZS'  there  is  given   PZ,  PS,  and  the 
angle  PZS.     Observe  that  PSS'  is  an  isosceles  A- 

Describe  the  meridian  PB  bisectincr  the  anole  S'PS,  and  then 
we  have  three  right  angled  spherical  triangles,  BPS,  BPS\  and 
BPZ  ;  taking  the  last,  we  have  the  following  proportion  : 

R     :     sin.  PZ    :    :     sin.  PZB     :     sin.  PB 
That  is,  P  :  cos.  16°  40'  :  :  sin.  68°  30'  :  sin.  P^=sin.63°  2' 30". 
To  find  ZB,  we  take  the  following  proportion,  (see  page  185, 
observation  1,  Robinson's  Geometry)  : 

P     :     coB.ZB     :    :     cos.  BP     :     cos.  PZ 
That  is,     P     :     cos.  Z^     :    :     cos.  63°  2' 30"     :     sin.  16°  40' 

P  sin.  16°  40' 19.457584 

cos.  63°  2'  30" 9.656411 

cos.  50°  45'  48" 9.801173 

To  find  S'B,  we  have 

P     :     COS.  S'B     :    :     cos.  63°  2' 30"     :     sin.  23°  18' 

P  sin.  23°  18' 19.597196 

COS.  63°  2'  30" 9.656411 

•  COS.  29°  14'  38" 9.940786 

Observe  that  S'B=:BS ;  therefore,  Z;S'=50°  45'  48"+29°  u' 
38"=80°  0'  26",  and  Z>S"=50°  45'  48"— 29°  14'  38"=21°  31'  10", 
the  complement  of  the  altitudes.  Consequently  the  altitude  at 
the  first  observation  was  9°  59'  34",  and  at  the  second,  68°28'50".» 

*  Our  results  differ  a  little  from  the  given  answer,  owing,  perhaps,  to  our 
not  being  minute  in  taking  out  the  logarithms,  or  finding  the  nearest  second 
corresponding  to  a  given  logarithm. — Experienced  men  on  these  matters  do 
not  pi'etend  to  work  to  seconds. 


TRIGONOMETRY. 


285 


To  find  the  time  from  noon  at  the  first  observation,  we  have  the 
following  proportion  : 

sin.  PaS^    :     sin-PZ^S^    :  ":     s'm.  ZS     :     sin.  ZFS.      That  is, 
cos.23°  18'  :  sin.68°30'  :  :  sin.80°  0'  26"  :  sin.ZPAS^=sin.86°5'30" 

Had  the  angle  been  90°,  the  time  would  have  been  just  6h.  but 
the  angle  3°  54'  30"  less ;  this  corresponds  to  15m.  38s.  in  time. 
Therefore,  the  time  was  6h.  15m.  38s.    For  the  time  at  the  second 
observation,  we  have 
cos.23°18'  :  sin.68°30' : :  sin,21°31'10"  :  sin.ZPAS"=sin.21°48'40" 

21°  48'  40"=lh.  31m.  14s.  from  noon,  or  lOh.  32m.  46s.  ap- 
parent time  in  the  morning. 

(10.)  An  observer  in  north  latitude  marked  the  time  when  the  stars 
Megulus  and  Sjnca  were  eclipsed  by  a  plumb  line,  that  is,  they  were 
both  in  the  same  vertical  plane  passing  through  the  zenith  of  the  ob- 
server. One  hour  and  ten  minutes  afterwards,  Regulus  was  on  the 
observer's  meridian.      What  was  the  observer's  latitude  ? 

The  positions  of  the  stars  in  the  heavens  were 

Regulus,  right  ascension  lOh.    Om.  10s.     Dec.  12°  43'       north. 

Spica,         *'  "      13h.  17m.    2s.     Dec.  10°  21' 20"  south. 

Let  R  be  the  position  of  Regu-  ' 
lus,  S  the  position  of  Spica,  P  the 
pole,  and  Z  the  zenith. 

Then  the  side  PaS'=100°21'20", 
PE=n°  17',  and  the  angle  BPS 
=3h.  16m.  52s.,  converted  into  de- 
grees ;  that  is,  PPS=49'^  13'. 

One  hour  and  ten  minutes  re- 
duced to  arc,  give  17°  30';  but  the 
stars  revolve  according  to  siderial, 
not  solar  time,  and  to  reduce  solar 
to  siderial  arc  we  must  increase  it  by  about  its  ^\-g  th  part ;  this 
gives  about  3'  to  add  to  17°  30',  making  17°  33'  for  the  angle 
ZPR.  Our  ultimate  object  is  to  find  PZ,  the  complement  of 
the  latitude. 

In  the  A  PRS,  we  have  the  two  sides  PB,  PS,  and  the  in  • 
eluded  angle  P,  from  which  we  must  find  PS  and  the  angle  SEP, 
and  we  can  let  a  perpendicular  fall  from  M  on  to  the  side  PS  and 


286  ROBINSON'S  SEQUEL. 

solve  it  in  the  usual  way  ;  but  to  show  that   a  wide  field  is  open 

for  a  bold  operator  ;  we    will  put  the  unknown  arc  IiS=x,  the 

side  opposite  Ii=r,  and  opposite  S=Sy  and  apply  one  of  the 

equations  in  formula  (S),  page  191,  Robinson's  Geometry. 

rpr    -  •  D     cos.a; — cos.r  cos.s 

That  IS,  cos.i^= 

sm.  r  sin.5 

Whence,   cos.P  sin.r  sin.s-j-cos.r  cos.5=cos.a; 

We  now  apply  this  equation,  recollecting  that  radius   is  unity, 

which  will  require  us  to  diminish  indices  of  the  logarithms  by  10. 

cos.P=cos.  49°  13' —1.815046 

sin.r=sin.lOO°  21' 20". .  .—1.992068       —cos —1.254579* 

sin.  s  =sin.77°  17' —1^89214  cos —1.342679 

0.6268 —1.797123      .03956 . .  .—2.597258 

cos.a;=0.6268— 0.03956=. 58724. 
Whence,  by  the  table  of  natural  cosines,  we  find  a:=54°2'20". 
To  find  the  angle  SEP  or  ZEP,  we  have 
sin.  54°  2'  20"  :  sin.  49°  13'  :  :  sin.  100°  21'  20"  :  sin.  ZBP 
Whence,     ZEP=66°  57'  30". 

Let  fall  the  perpendicular  PB  on  PZ  produced,  then  the  right 
angled  spherical  A  PPP  gives  this  proportion  : 

P  :  sin.  77°  17'  :  :  sin.  17°  33'  :  sm.PB=sm.  17°  6'  22" 
To  find  PP  we  have 

P  :  COS.  PB  :  :  cos.  17°  6'  22"  :  cos.  77°  17' 
Whence,  P^=76°  41'.     Now  to  find  the  angle  BPP,  we  have 

sin.  77°  17'  :  P  :  :  sin.  76°  41'  :   sin.  ^i2P=sin.  86°  1' 
From  PPB  take  PPZ,  and  ZPB  will  remain  ;  that  is. 
From  86°  1'  take  66°  57'  30",  and  ZPB=19°  3'  30". 
By  the  application  of  equation  (12),  we  ^nd  that 
P  sin.  17°  6'  22"=tan.  BZ  cot.  19°  3'  30" 
Whence,  ^Z=5°  48'     And  PZ=76°  41'— 5°  48'=70°  63'. 
The  complement  of  70°  53'  is  19°  7',  the  latitude  sought. 
By  this  example  we  perceive  that  by  the  means  of  a  meridian 
line,  a  good  watch,  and  a  plumb  line,  any  person  having  a  knowl- 
edge of  spherical  trigonometry,  and  having  a  catalogue  of  the  stars 
at  hand,  can  determine  his  latitude  by  observation. 

♦Observe  that  r  is  greater  than  90®,  its  cosine  is  therefore,  negative  in  value, 
rendering  the  product  cos.  r  cos. «,  or  .03956,  negative. 


PART  FOURTH. 

PHYSICAL.  ASTRONOIWY. 

KEPLER'S  LAWS. 

1 .  The  orbits  of  the  planets  are  ellipses,  of  which  the  sun  occupies 
one  of  the  foci. 

2.  The  radius  vector  in  each  case  describes  areas  about  the  focus 
which  are  proportional  to  the  times. 

3.  The  squares  of  the  times  of  revolution  are  to  each  other  as  the 
cuhes  of  the  mean  distances  from  the  sun. 

The  first  of  these  is  a  mere  fact  drawn  from  observation.  The 
second  is  also  an  observed  fact — but  susceptible  of  mathematical 
demonstration,  under  strict  geometrical  principles,  and  the  law  of 
inertia.  The  demonstration  is  to  be  found  in  Robinson's  Astron- 
omy, and  in  various  philosophical  works. 

The  third  is  also  susceptible  of  demonstration  by  means  of  the 
calculus — and  by  simple  geometrical  proportion,  if  we  suppose 
the  orbits  circular. 

We  now  propose  to  investigate  and  determine  the  relative  times 
of  revolutions  of  two  bodies  about  the  sun,  on  the  supposition 
that  they  revolve  in  circles,  (which  is  not  far  from  the  truth,)  and 
are  attracted  towards  the  center  inversely  proportional  to  the 
squares  of  their  distances. 

Let  S  be  the  center  of  the  sun,  AS 
the  radius  vector  of  one  planet,  and  SV 
that  of  another. 

Let  m  be  the  mass  of  the  sun,  SA=^ri 

and  S  V=E.     Then  ~  is  the  force  which 

is  exerted  on  the  planet  at  A,  and  -—  is 

the  force  exerted  on  the  other  planet  at  V. 

If  we  take  any  small  interval  of  time, 

say  one  minute,  and  let  AI)  represent  the 

distance  the  first  planet  falls  from  the  tan- 


288  ROBINSON'S   SEQUEL. 

gent  of  its  orbit  in  unity  of  time,  and  VH  the  distances  the  other 
falls  in  the  same  time, 

Then        ^     ;     ^     '.    '.     AD     '.     VH  (1) 

That  the  planets  may  maintain  themselves  in  their  orbits,  the 
first  must  run  over  the  arc  AB  in  the  unit  of  time,  and  the  sec- 
ond must  run  over  the  arc  VF.  But  this  interval  or  unit  of  time 
can  be  taken  ever  so  short ;  and  when  very  short,  as  a  minute  or 
a  second,  AB  and  VF,  may,  yea  must  be  considered  straight  lines, 
chords 'Comc'iding  with  the  arc. 

But  if  we  take  any  chord  of  an  arc,  as  AB,  and  from  one  ex- 
tremity draw  the  diameter,  and  from  the  other  let  fall  the  perpen- 
dicular BD,  we  shall  have 

AB     :     AB     :    :    AB     :     2r 

AB^                                             VF^ 
Whence,       AI)= ,  and  in  like  manner  VIf= 

Substituting  these  equals  in  proportion  (1  j,  and  dividing  the 
first  couplet  by  m,  and  multiplying  the  last  couplet  by  2,  we  have 

1  1  AB^     .      VF^ 


W  r  R 


1  1 


Or,  _     :     _     :    :     AB^     :     VF^  (2) 

r  R 

Because  the  first  planet  is  supposed  to  run  along  the  arc  AB, 

in  one  minute,  the  number  of  minutes  it  will  require  to  make  its 

revolution  will  be  found  by  dividing  the  whole  circumference  by 

AB.     The  circumference  is  expressed  by  2r7t,  and  put  t  to  repre- 

sent  the  time  of  revolution  ;   then  t  = ,   or  AB  =  - —     In 

AB  t 

the  same   manner  if  T  represents  the  time  of  revolution  of  the 

272  Tt 
second  planet,  we  must  have  VF=^——-.     By  squaring  these  ex- 
pressions and  substituting  the  values  of  AB^  and  VF^  in  pro- 
portion (2),  we  have 


r 


R  t^  T^ 


O  1    •    JL    •  •    l!    •    :?" 


ASTRONOMY. 

Multiply  the  first  couplet  by  tR,  then 
i2     :    r     :    .     l!     : 


Or, 


R^ 


R^ 

1^2 


L.=.^,    Whence,  t^  :  T"  : 


289 


•3   :  i23. 


This  last  proportion  corresponds  with  Kepler's  third  law. 


The  following  propositions  are  to  be  found  on  page  146  of 
Robinson's  Astronomy.  The  frequent  requests  we  have  received 
to  demonstrate  them,  suggested  the  propriety  of  pubhshing  the 
demonstrations  in  this  connection. 

The  propositions  are  as  follows  : 

(1.)  If  two  comets  m-ove  in  parabolic  orbits,  the  areas  described  by 
them  in  the  same  time  are  proportional  to  the  square  roots  of  their 
perihelion  distances. 

Conceive  a  comet  to  revolve  in  an 
ellipse,  F'  the  position  of  the  sun,  and 
A'F'  the  perihelion  distance. 

Let  F'D=r,  F'C=x,  DC=y,  and 
put  t  to  represent  the  number  of  hours 
required  by  the  comet  to  make  a  rev- 
olution. 

Now  nry  ==  the  area  of  the  ellipse.  This  area  divided  by  t, 
will  express  the  area  described  by  the  comet  about  the  sun  in  one 
hour.     Let  that  area  be  represented  by  a. 


Then 


nry  _ 


a.      Let  R,  x\  y\  T,  and  A,  represent  similar 
quantities  pertaining  to  another  orbit,  and  by  parity  of  reasoning, 

T 

Whence, 

By  squaring, 


ry^ 
t 
r^y^ 


T      '    ' 

qi3 


By  Kepler's  3d  law,  t^   :  T^ 
19 


:  R\  or  t^  = 


(1) 


290  ROBINSON'S  SEQUEL. 

The  value  of  /'  substituted  in  (1),  and  reduced,  will  give 

y-     X    y       :    :     a^     :    A^  (2) 

r  xt 

By  inspecting  the  right  angled  triangle  F'CDy  we  readily  per- 
ceive that  y^=:r^ — x'=(r-\-x)  (r — x).     Similarly,  y'^=(B-\-x') 

Now  if  we  suppose  the  ellipse  to  be  infinitely  eccentric,  (as  we 
must  when  it  becomes  a  parabola,)  {r-\'x)  =  2r  nearly,  and 
(r — x)^=A'F'z=p  exactly,  (calling  p  the  perihelion  distance  of 
one  comet,  and  P  the  perihelion  distance  of  the  other.) 

Similarly  ,  {R+x')z=9,R,  and  (B—x')=F. 

Substituting  these  values  in  (2),  we  have 

r  R 

Or,  p     '.       P        \    \     a^     \     A^ 

Or,  Jp     :      JP      :    :     a       :     A         Q.  E.  D. 


,  (2.)  j^we  sui^pose  a  planet  moving  in  a  circular  orhit,  whose  radius 
is  equal  to  the  perihelion  distance  of  a  cornet  moving  in  a  parabola, 
the  areas  described  by  these  two  bodies,  in  the  same  time,  will  be  to 
each  other  as  1  to  the  square  root  of  2.  Thus  are  the  motions  of 
comets  and  planets  cminected. 

Let  S  be  the  position  of 
the  sun,  P  the  perihelion  point 
of  a  comet  revolving  in  an 
ellipse. 

Put  SP=x,  and  let  i=the 
time  in  which  the  planet 
would  revolve  in  the  circle, 
and  T=  the  time  required 
for  the  comet  to  revolve  in 
the  ellipse. 

By  the  first  law  of  Kepler  the  same  body  describes  equal  areas 
in  equal  times  ;  therefore  if  we  divide  the  area  of  the  circle  by 
the  number  of  units  in  the  time  of  revolution,  we  shall  have  the 
area  described  in  one  unit  of  time. 

The  area  of  the  circle  is  ^ar*,  and  this  divided  by  <,  gives 


ASTRONOMY.  291 

=  the  sector  described  by  the  planet  in  unity  of  time.     Also, 

I 

=  the  sector  described  by  the  comet  in  the  same  time. 

Conceive  these  two  sectors  to  commence  on  the  line  SP,  then 

3  A   Ti 

(sector  in  circle)  :  (sector  in  ellipse)   :  :  —  :  -__        (1) 

f  JL 

A  and  B  are  the  semi-conjugate  axes  of  the  ellipse. 

By  Kepler's  third  law, 

e     :     T^     :    :    x^     :     A^  (2) 

Multiplying  the  last  couplet  of  ( 1 )  by  tT,  gives 

(sector  in  circle)  :  (sector  in  ellipse)   :  :   Tx^   :  [tA)B 

By  squaring,  we  have 
(sec.incircle)^  :  (sec.  in  ellipse  )2  ::  {T''x^)x  :  (il^^^^^a     ^3^ 

From  (2)  we  find  T^x^=.t^A^,  and  substituting  the  value  of 
T^x^  in  (3),  we  have 
(sec.incircle)2  :  (sec.inellipse)2  ::    t^'A^'x    :    (^M^)^^^ 

:\       Ax       \        B^  (5) 

Observe  the  right  angled  triangle  CSQ.    SG=A,  CS=A — x, 
OG=B. 

A''—(A—xy=B^ 
Or,  2JaJ— a;2=52 

Substituting  this  value  of  B^    in   (5),    and  dividing  the  last 
couplet,  gives 

(sector  in  circle)^  :  (sector  in  ellipse)  ^  :  :  A  :  2A — x 

Dividing  the  last  couplet  by  A,  and  extracting  the   square 
root,  gives 

(sector  in  circle)  :  (sector  in  ellipse)  :  :  1  :  ^2 — —  (6) 

When  the  ellipse  is  very  eccentric,  A  is  very  great  in  relation 

,  X    • 

to  X,  and  the  fraction,  —  is  then  very  insignificant  in  value.     As 

an  ellipse  becomes  more  and  more  eccentric,  its  curve  approaches 
nearer  and  nearer  to  a.  parabola,  and  when  it  becomes  a  parabola, 

A  is  infinite  in  respect  to  x,  and  the  fraction  —  is  then  absolutely 

zero,  and  proportion  (6)  becomes 

(sector  in  circle)  :  (sector  in  parabola)  :  :  I  :  J2      Q.  E.  D. 


292  ROBINSON'S  SEQUEL. 

The  following  inquiry  has  frequently  come  to  us.  We  now 
give  it  in  the  words  of  a  correspondent. 

Mr.  Robixsox  : 

Dkar  Sir. — On  page  192,  Art-  180  of  your  Astronomy,  it  is  stated  that 
because  the  mean  radial  force  causes  the  moon  to  circulate  at  -I.-  part 
greater  distance  from  the  earth  than  it  otherwise  would,  its  periodical  revo- 
lution is  increased  by  its  179th  part.  The  question  is,  where  does  the  fraction 
-K    come    from? 

RE  PL  y. 

The  mean  radial  force  acting  in  the  direction  of  the  radius 
vector  does  not  prevent  the  moon  from  describing  equal  areas  in 
equal  times.  Therefore  the  moon  describes  the  same  area  with, 
as  it  would  without  this  action  ;  but  the  radius  is  increased,  and 
consequently  the  angular  velocity  diminished. 

We  will  now  give  the  increase  of  radius,  and  require  the  cor- 
responding decrease  of  angular  velocity,  and  we  shall  find  the 
ratio  of  one  will  be  double  that  of  the  other,  on  the  condition  that 
the  increase  or  decrease  of  either,  is  small  in  relation  to  the  whole- 

Let  U  be  the  angular 
point  of  two  equal  sectors, 
7'  the  radius  of  one,  and  A 
its  arc.  x  its  angle  on  the 
radius  of  unity. 

Let  {r-\-h)  be  the  radius 
of  the  other  sector,  A  ^  its  arc 
and  y  its  angle. 

Then  by  reason  of  the  two 
equal  sectors,  rA=(r-{-h)A^  (1) 

From  one  sector,     \   :  x  :  :  r  :  A.     Or,  A=rx. 

From  the  other,       1   :  y  :  :  (r-\-h)  :  A^     Or,  A ^=(r-\-h)y. 

Substituting  the  values  of  A  and  ^,  in  (1),  we  have 
r'x={r-\-kyj/ 

Or,  X     :     y     :    :     (r-]-hy     :     r^ 

Or,  %     \    y     \    \     r-^-'lrh-^-h^     :     r^ 

Because  A  is  a  very  small   fraction  in  relation  to  r,  h^  can  be 

omitted ;  then 

X     \     y     \    \     r^-f-2rA     :     r^ 

Or,  X     \    y     \    '.     r  -j-2  h          r 


ASTRONOMY.  293 

This  last  proportion  shows  that  if  the  radius  r  is  increased  by  h, 
the  angular  velocity  and  consequently  the  periodic  time  must  be  di- 
minished by  2A. 

PROPOSITION . 

Given  the  position  of  the  earth  as  seen  from  the  sun,  the  position 
of  any  other  planet  as  seen  from  the  sun,  to  find  the  position  of  that 
planet  as  seen  from  the  earth. 

The  motion  of  the  earth  and  planets  being  known,  and  the 

elements  of  their  orbits,  the  astronomical  tables  give  the  position 

of  the  earth  and  any  planet  for  any  given  instant  of  time..    The 

position  of  the  planet  from  the  earth  must  then  be  computed  by 

plane  trigonometry.      But  before  we  give  a  definite  example,  we 

adduce  the  following 

LEMMA. 

1 .  In  any  plane  triangle  the  greater  of  two  sides  is  to  the  less,  as 
radius  to  the  tangerd  of  a  certain  angle. 

2.  Radius  is  to  the  tangent  of  the  difference  between  this  angle  and 
46°,  as  the  tangent  of  half  the  sum  of  the  angles  at  the  base  of  the  tri- 
angle is  to  the  tangent  of  half  their  difference. 

To  obtain  that  certain  angle,  we  must  place  the  two  sides  at 
right  angles  to  each  other. 

Let  CA  be  the  greater  of  two 
sides  of  a  A,  and  CE  a  less 
side  placed  at  right  angles ;  then 
CAE  is  the  certain  angle  spoken 
of,  less  than  45°,  and  EAB  is 
the  difference  between  it  and 
45°. 

From  (7  as  a  center  with  the 
longer  side  as  radius,  describe  the  semicircle.  Then  DE  =  the 
sum  of  the  sides,  and  EG  their  difference.  Join  DA,  A  G,  and 
from  E  draw  EB  parallel  to  DA.  DA  G  is  &  right  angle  because 
it  is  in  a  semicircle  ;  therefore,  EB  being  parallel  to  DA;  EBG 
is  a  right  angle  also.     DA=zAG,  and  EBz=BG. 

Let  a  be  the  greater  side  of  a  triangle  represented  in  magnitude 
but  not  in  position,  by  CA,  and  c  the  shorter  side,  represented  in 
magnitude  by  CE ;  then  it  is  obvious  that 

a     '.     c     '.  '.     R     :     tan.  CAE 


■ 


294 


ROBINSON'S  SEQUEL. 


This  angle  taken  from  the  table  and  subtracted  from  46°  will 
give  the  angle  EAB.     By  proportional  triangles  we  have 


BE 
That  is,  a-f-c 
But.  AB 

Whence,   a-\-c 


EG 


EB 


AB 
AB 

B 

R 


BG^EB 
EB 

tan.  EAB 
tan.  EAB 


By  proportion  7,  page  149,  Robinson's  Geometry,  we  find  that 
a-^c  :  a — c  : :  tan.^sum  ang.  atbase  :  tan.  ^  their  diflf. 

Therefore  by  comparison, 
R  :  tan.-£'^-5  ::  tan. | sum  ang.  at  base  :  tan.^ their  difF.  Q.E.D. 

The  application  of  this  proposition  is  very  advantageous  when 
the  logarithms  of  the  two  sides  of  a  triangle  are  given  and  not 
the  sides  themselves.  It  obviates  the  necessity  of  finding  the 
numerical  values  of  the  sides.  This  proposition  is  almost  solely 
used  in  Astronomy,  and  we  give  the  following  example  as  an 
illustration. 

In  the  Nautical  Almanac  for  1864,  Ifirid  that  on  the  first  day  of 
April  at  noon,  mean  time  at  Greenwich,  the  sun's  longitude  is  11° 
26'  28",  and  the  logarithm  of  the  radius  vector  of  the  earth  is 
0.0000224.  At  the  same  time  the  heliocerUric  longitude  of  Jupiter 
is  283°  46'  7",  soitth  latitude  6'  41",  and  logarithm  of  its  radius 
vector  0. 71 45152.  Required  the  geoceyitric  latitude  and  longitude 
of  Jupiter,  and  the  logarithm  of  its  true  distance  from  the  earth. 

Let  S  be  the  sun,  'Y'=a=  the  line  made  by 
Aries  and  Libra  in  the  plane  of  the  ecliptic, 
/y^  <fcc.,  the  direction  of  counting  lon- 
gitude. 

Place  the  earth  at  E,  so  that  the  sun  at 
S  will  appear  to  be  in  11°  26'  28"  of  longi- 
tude. Then  E,  the  earth,  will  appear  from 
the  sun  to  be  in  191°  26'  28"  of  longitude. 
jS'jB'  is  a  very  little  over  a  unit  in  distance, 
as  we  see  by  the  log.  0.0000224. 

The  longitude  of  Jupiter  as  seen  from  the 
sun  is  283°  46'  7" ;  hence,  draw  SI  so  that 
the  angle  JD=^/will  be  103°  46'  7",  and  its  distance  from  S  to  I 
a  little  over  S  times  SE.  The  angle  ESI  will  be  92°  19'  39", 
and  log.  of  SI  is  given  at  0.7145152.     Our  object  is  to  find  the 


ASTRONOMY.  9U 

position  of  the  line  BI,  or  Sh  which  is  supposed  parallel  to  HI, 
and  the  logarithm  of  the  distance  HI. 

Jupiter  not  being  in  the  plane  of  the  ecliptic,  we  must  reduce  it 
to  that  plane,  by  multiplying  its  distance  by  the  cosine  of  its 
inclination. 

Thus,  to  the  log 0.7145152 

Add  cos.  6'  41" 9.9999998 

Log.  of  distance  in  the  ecliptic, . .  .^ 0.7145150 

Now  by  the  first  part  of  the  Lemma, 

As 0.7145150 

To 0.0000224 

So  is  radius 10:0000000 

,To  tan.  10°  55'  21" 9.2855074 

This  arc  from  45°  gives  34°  4'  39".  The  angle  U SI  irom  180° 
gives  87°  40'  21"  for  the  sum  of  the  angles  U  and  /.  Their  half 
sum  is  therefore,  43°  50'  10"5.  Let  their  half  difference  be  de- 
noted by  X.     Then  by  the  last  part  of  the  Lemma, 

B     :     tan.  34°  4'  40"     :    :     tan.  43°  50'  10"5     :     tan.  x 

tan.  34°  4'  39" 9.830254 

tan.  43°  50'  10"5 9.982352 

tan.a:  33°  0'  19" 9.812606 

The  angle  ^=43°  50'  10"+33°  0'  19"=76°  50'  29". 
The  angle  7=43°  50'  10"— 33°  0'  19"=10°  49'  51". 
The  angle  ISk=\0°  49'  51";  therefore,  the  geocentric  loftgi' 
tude  of  Jupiter  is  283°  46'  7"+10°  49'  51"  or  294°  35'  58". 
For  the  log.  of  JSI^re  have  the  following  proportion  : 

sin./    :     SB    :    :     sin.  ISU    :     UI 
Or,  sin.  10°  49' 51"  :  SU  :    :     sin.  92°  19'  39"  :  £1 

Log.  sin.  92°  19'  39" 9.999641 

Log.  SU 0.0000224 

9.9996634 

Log.  sin.  10°  49-51" 9.2739400 

Log.  UI 0.7257234 

This  result  is  the  logarithm  of  the  distance  from  U  along  in 
the  plane  of  the  ecliptic  to  the  point  where  the  perpendicular  falls 
from  Jupiter, — the  hypotenusal  or  absolute  distance  is  a  little 


296  ROBINSON'S  SEQUEL. 

greater,  but  it  is  hardly  perceptible  in  this  case,  as  Jupiter  is  so 
near  the  ecliptic.  Indeed  it  would  increase  the  last  decimal  fig- 
ure in  the  lo'garithm  by  2,  making  it  6. 

Jupiter  appears  from  the  sun  at  this  time  to  be  6'  41"  south  of 
the  ecliptic,  but  from  the  earth,  the  angle  between  it  and  the 
ecliptic  would  not  be  so  great,  because  EI  is  greater  than  SI. 
But  to  compute  the  geocentric  latitude  of  Jupiter  or  any  other 
planet  exactly,  we  have  the  following  principle  : 

We  refer  in  particular  to  this  example,  but  the  principle  is 
general. 

Conceive  the  perpendicular  distance  of  t\ie  planet  from  the 
ecliptic  to  be  represented  by  i>,  and  let  this  distance  be  made 
radius ;  then  IS  will  be  the  cotangent  of  the  heliocentric  latitude, 
and  IE  the  cotangent  of  the  geocentric  latitude. 

Denote  the  geocentric  latitude  by  x  ;  then 

D     \     R     \    \     IS    \     cot.  6' 41" 
And  D     :     E     '.    '.     IE    :     cot.  a; 

7-ET 

Whence,  IS  :  cot  6' 41"     :    :     IE   :  cot.  ar=lr  cot.  6' 41" 

That  is,  Erom  the  log.  of  the  ijlaneCs  distance  from  the  earth, 
svhtract  the  log.  of  its  distance  from  the  sun,  and  to  the  difference  add 
the  log.  cotangent  of  the  heliocentric  latitude,  and  the  sum  is  the  log. 
cot.  of  the  planet's  geocentric  latitude. 

To  apply  this  equation  with  accuracy,  requires  some  little  tact 
in  using  logarithms.  Observe  that  cot.  6'  41"  is  the  tan.  of  6'  41", 
subtracted  from  20.0000.  To  find  the  tan.  of  6'  41"  or  401",  first 
find  the  tangent  of  1",  fhen  add  the  log.  of  401. 

tan.  l'=60" *. .  .6.463726 

sub.  log.60 1.778151 

tan.  1" 4.685575 

log.  401 2.603144 

tan.  6'  41" 7.288719 

Log.  -£^/— log.  //S'=0.7257234— 0.7145150=0.0112084. 

The  log.  cot.  must  be  increased  by  this  quantity,  therefore  the 
log.  tan.  must  be  diminished  by  the  same  ;  hence 

7.2887190 
0.0112084 


ASTRONOMY.  '  29^7 

Log.  tan.  of  geocentric  latitude  is 7.2775106 

Subtract  log.  tan.  of  1" 4.685575 

Log.  of  390"8,  or  6'  31"  nearly, 2.5919356 

Thus  we  find  the  geocentric  latitude  of  Jupiter  to  be  6'  31" 
south  at  this  particular  time. 

Having  the  planet's  latitude  and  longitude,  we  can  compute  its 
corresponding  right  ascension  and  declination,  and  the  following 
results  will  be  obtained  : 

Right  ascension,  19h.  46m.  9s.     South  declination  21°  19'  41". 

SOLAR  ECLIPSES. 

We  will  now  show  the  computation  to  determine  the  times  of 
beginning  and  end,  and  other  circumstances  attending  a  solar 
eclipse  as  seen  from  any  assumed  locality  on  the  earth.  No  person 
can  do  this  with  any  safety,  depending  on  the  rules  of  another, 
he  must  understand  the  nature  and  scope  of  the  problem  for  him- 
self. It  requires  a  general  knowledge  of  astronomy  and  philoso- 
phy, and  a  familiar  knowledge  of  both  plane  and  spherical 
trigonometry. 

The  mathematical  philosophy  of  the  subject  is  explained  gen- 
erally on  page  214  of  Robinson's  Geometry,  and  here  we  will 
illustrate  it  by  an  example. 

As  near  as  we  can  determine  by  some  rough  projections,  the 
eclipse  of  May  26,  1854,  will*  be  nearly  central  and  annular  as 
seen  from  Burlington  in  Vermont.  Curiosity  has,  therefore,  led 
us  to  make  minute  calculations  for  that  place. 

We  take  the  elements  from  the  English  Nautical  Almanac. — 
Let  the  reader  observe  that  the  elements  here  correspond  to  the 
mean  time  of  conjunction  in  rif/ht  ascension.  The  elements  in 
Robinson's  Astronomy  correspond  to  conjunction  in  longitude,  the 
difference  is  8m.  37s.  in  time. 

1854,  May,  26. 

Greenwich  mean  time  (/  in  R.  A 8h.  55m.  43.8s. 

Sun  and  moon's  R.  A * 4h.  13m.  7.41s. 

Moon's  Dechnation  North, 21°  33'  3r'8 

*  This  was  written  in  April,  1853,  and  therefore  spoken  of  in  the  future 
tense. 


«98 


ROBINSON'S  SEQUEL. 


Sun's  Declination  North, 21°  11'  16"8 

Moon's  Horary  motion  in  R.  A 31'  18"9 

Sun's  Horary  motion  in  R.  A 2'  31"8 

Moon's  Horary  motion  in  Declination  N 8'    7"3 

Sun's  Horary  motion  in  Declination  N 25"9 

Moon's  Equatorial  Horizontal  parallax 54'  32"6 

Sun's  **  **  **         8"5 

Moon's  semidiameter  14'  53"5.     Sun's  S.  D.   15'  48"9. 
Lat.  of  Burlington  44°  28'  N.     West  Long.  73°  14'=4h.  52m.  56s 

Greenwich  mean  time  of  q^ ^^'  55ra.  44s. 

Long,  in  time 4h.  52m.  66s. 


Mean  time  of  (/  at  Burlington . 
Equation  of  time,  add 


4h.    2m.  48s.  P.M. 
3m.  15s. 


Conjunction  at  B.,  apparent  time. . .  .4h.    6m.    3s.=61°  30'  45". 

As  the  earth  is  not  a  perfect  sphere,  (the  equatorial  diameter 
being  the  largest,)  the  equatorial  horizontal  parallax  requires  re- 
duction for  other  latitudes,  and  latitude  itself  requires  a  reduction 
at  all  points,  except  at  the  equator  and  the  poles. 

The  horizontal  semidiameter  of  the  moon  requires  .augmenta- 
tion, as  the  moon  rises  in  altitude,  for  the  nearer  the  moon  is  to 
the  zenith  of  the  observer,  the  nearer  it  is  in  absolute  distance. 

The  following  tables  correct  the  elements  in  these  particulars* 

Reduction  of  the  Parallax  and  also  of  the  Latitude. 


Lat. 

Red. 
of  par. 

Red.  of 
Lat. 

Lat. 

Red. 
of  par. 

Red.  of 
Lat. 

Lat. 

Red. 
of  par. 

Red.  of 
Lat. 

o 

" 

/      // 

o 

"      1 

/      II 

o 

/' 

/      II 

0 

0.0 

0    0.0 

3 

0.0 

1  11.8 

33 

3.3 

10  28.3 

63 

8.8 

9  18.3 

6 

0.1 

2  22.7 

36 

3.8 

10  64.3 

66 

9.2 

8  32.9 

9 

0.3 

3  32.1 

39 

4.4 

11  13.2 

69 

9.7 

7  42.0 

12 

0.5 

4  39.3, 

42 

4.9 

11  24.7 

72 

10.0 

6  45.9 

16 

0.7 

6  43.4 

46 

6.5 

1128.7 

75 

10.3 

6  46.4 

18 

1.0 

6  43.7 

48 

6.1 

1125.2 

78 

10.6 

4  41.0 

21 

1.4 

7  39.7 

51 

6.7 

11  14.1 

81 

10.8 

3  33.5 

24 

1.8 

8  30.7 

54 

7.2 

10  65.7 

84 

11.0 

2  23.7 

27 

2.3 

9  16.1 

67 

7.8 

10  30.0 

87 

11.1 

1  12.3 

30 

2.7 

9  55.4 

60 

8.3 

9  57.4 

90 

11.1 

0    0.0 

ASTRONOMY. 


299 


Atigmentaiion  of  the  Mooti's  Semi- diameter. 

Horizon.  Semi-diameter. 

Alt. 

Horizon.    Semi-diameter. 

Alt. 

f4'30" 
If 

16' 

16' 
It 

17' 

14'30" 

15' 

16' 

It 

\r 

— OT 

H 

o 

// 

// 

2 

0.6 

0.6 

0.7 

0.8 

42 

9.2 

9.8 

11.2 

12.6 

4 

1.0 

1.1 

1.3 

1.5 

45 

9.7 

10.4 

11.8 

13.3 

6 

1.6 

1.6 

1.9 

2.1 

48 

10.2 

10.9 

12.4 

14.0 

8 

2.0 

2.1 

2.4 

2.7 

51 

10.6 

11.4 

13.0 

14.7 

JO 

2.4 

2.6 

3.0 

3.4 

54 

11.1 

11.8 

13.5 

15.2 

12 

2.9 

3.1 

3.6 

4.0 

57 

11.5 

12.3 

14.0 

15.8 

14 

3.4 

3.6 

4.1 

4.7 

60 

11.8 

12.7 

14.4 

16.3 

16 

3.8 

4.1 

4.7 

5.3 

63 

12.2 

13.0 

14.9 

16.8 

18 

4.3 

4.6 

5.2 

5.9 

<o^ 

12.5 

13.4 

15.2 

17.2 

21 

4.9 

5.3 

6.0 

6.8 

69 

12.8 

13.7 

15.6 

17.6 

24 

SQ 

6.0 

6.8 

7.7 

72 

13.0 

13.9 

15.9 

17.9 

27 

6.2 

6.7 

7.6 

8.6 

75 

13.2 

14.1 

16.1 

18.2 

30 

6.9 

7.3 

8.4 

9.5 

78 

13.4 

14.3 

16.3 

18.4 

33 

7.5 

8.0 

9.1 

10.3 

81 

13.5 

14.4 

16.5 

18.6 

36 

8.1 

8.6 

9.5 

11.1 

84 

13.6 

14.5 

16.6 

18.7 

39 

8.6 

9.2 

10.5 

11.9 

90 

13.7 

14.6 

16.7 

18.8 

Equatorial  horizontal  parallax 54'  32"6 

Reduction  for  latitude 6"4 

Reduced  h.  p 64'  27"2 

Subtract  sun's  h.  p 8"6 

Relative  or  effective  h.  p 64'  18"7=3268"7 

Lat.  of  Burlington 44°  28' 

Reduction 11'  27      ^ 

Reduced  latitude 44°  16'  33" 

As  the  sun  and  moon  are  west  of  the  meridian,  therefore  the 
moon  is  apparently  thrown  back  by  the  effects  of  parallax,  and 
consequently  the  beginning  of  the  eclipse  will  not  take  place  un- 
til about,  or  after,  the  time  of  conjunction. 

To  decide  this  point,  let  m  represent  the  place  of  the  moon 
at  4h.  6m.  3s.,  and  S  the  place  of  the  sun  at  the  same  time. 
^m=22' 15",  their  difference  in  declination,  mn  the  parallax  in 
altitude,  and  n  will  be  the  apparent  place  of  the  moon.      Our 


300 


ROBINSON'S  SEQUEL. 


object   is  to  find    the  dis- 
tance  between   S    and   n, 

and  we   accomplish  it   by 

the   aid   of    the    spherical 

triangle  FZ?n*.     We  have 

PZ,    Fm,    and    the    angle 

ZPm.     We  must  find  Zm 

and  the  angle  ZmP. 

We   use   the    following 

equation   taken  from  page 

209,  Geometry,  in  which  A 

represents  the  moon's  alti- 
tude, L  the  latitude  of  the 

place,  and  D  the  moon's  polar  distance. 

p_sin.  A — sin.  Zcos.  D 
cos.  L  sin.  J) 
Whence,  sin.  ^=cos.  Zm  =  cos.  P  cos.  L  sin.  D-\-sin.  L  cos.  i), 

cos.  P=cos.  61°  30'  45" 9.678489 

cos.  Z=cos.  44°  16'  33" 9.854910         sin 9.843917 

sin.  i>=sin.  68°  26'  28". ..  ..9.968853         cos 9.562944 

0.31787 —1.502252    .255192  —1.406861 

Whence,  the  natural  sine  of  the  moon's  true   altitude,  or  cos. 

Zm=0.31787+0.25519=.57306. 

*  As  the  moon  is  at  m,  and  the  sun  at  <S'.  on  the  same  meridian  PjS,  which  is 
61°  30'  45"  west  of  Burlington,  or  134°  44'  45"  west  of  Greenwich;  there- 
fore, this  is  the  meridian  on  which  the  sun  will  be  centrally  eclipsed  at  appa- 
rent noon  ;  and  the  latitude  will  be  such  that  mS  must  be  the  moon's  parallax 
in  altitude.  To  find  that  latitude  is  a  very  easy  and  interesting  problem. 
Let  the  latitude  be  such  that  the  apparent  altitude  of  the  moon  shall  be  rep- 
resented by  X.     Then  3257"  cos.  ar=mS=1335".  (Rad.  1.) 

1335\R 13.125481 

cos.j;=_ 

3258.7 3.513044 

COS.  ar=sin.  moon's  app.  zenith  distance  24°  11*   5"  sin 9.612437 

Moon's  declination  north 21°  33'  32" 

Lat.  (apparent,) 45°  44'37" 

Reduction  of  Lat 11*29" 

True  Lat 45°  33'        north. 

Hence  the  sun  will  be  centrally  eclipsed  at  noon  in  longitude  134°  45'  west 

and  latitude  45°  33'  north. 


ASTRONOMY.  301 

By  taking  out  the  corresponding  arcs,  we  find 

Moon's  true  altitude=34°  58'  10"      Zm=55°  V  50". 
sin.  Zm     :     sin.  P     :    :     sin.  ZP     :     sin.  Z»iP=  Smn. 
Or,  sin.  55°  1' 50"  :  sin.  61°  30' 45"  ::  cos.44°  16'33"  :  sin. /Smn 
==sin.  60°  10'  8". 

The  next  and  most  delicate  step  is  to  obtain  mji,  the  parallax 
in  altitude. 

If  we  use  the  true  altitude  of  the  moon  for  the  apparent  alti- 
tude, we  can  find  the  approximate  value  of  mn  as  follows  :  (see 
page  201,  Surveying  and  Navigation.) 

Horizontal  parallax  3258"7         log 3.513044 

COS.  34°  58'  10"         add .9.913526 

1st  approx.  value  of  7nn  44'  30"=2670" 3.426570 

Moon's  true  alt 34°  58'  10" 

Sub.  parallax  in  alt. ...       44'  30" 
Moon's  appa.  alt.  nearly .  .34°  13'  40" 

H.  p.  as  before, , . .  .3.513044 

COS.  34°  13'  40" ,    9.917380 

2d  approx.  value  of  mn,  44'  54"=2694" 3.430424 

H.  p.  as  before 3.513044 

Moon's  appa.  alt cos.  34°  1 3'  1 6" 9.91 7422 

True  value  of  mn,  44'56"6=2694"6 .3.430466 

Now  in  the  A  Smn,  (which  we  may  conceive   to  be  a  plane 

triangle,)we  have  Sm=1335",  mn=2694"G,  and   the  angle  Smn, 

50°  10'  8",  to  find  Sn. 

If  from  n  we  conceive  a  perpendicular  to  be  let  fall  on  to  the 

meridian  Sm,  and  designate  it  by  p,  and  the  other  side  of  the  right 

angled  triangle  thus  formed  by  q,  then  we  shall  have 

H     :     2694.6     :    :     sin.  50°  10'  8"     :    ;> 
And         i?     :     2694.6     :    :     cos.  50°  10' 8"     :     q 

mn 3.430466 3.430466 

sin.  60°  10'  8" .9.885322     cos 9.806537 

p     2069.2 3.315788     q  1726.8. .  .3.237003~ 

Observe  that  p  is  the  effect  of  parallax  perpendicular  to  the 
lunar  meridian  at  that  time;  and  q  is  the  parallax  in  declination. 


302  ROBINSON'S   SEQUEL. 

Moon^s  true  declination  north  of  the  sun 1336" 

Moon's  parallax  in  declination  south 1725"8 

Moon's  apparent  dec.  south  of  the  sun 390^8 

The  apparent  distance  between  sun  and  moon  is,  therefore, 

V(2069.2)2+(390.8)2=2105"7 
Moon's  S.  D.,  14'  53"5.  Augmentation  for  altitude  8".  Sun's 
S.  D.  16'48"9.  Sum  =  30' 50"4=1860"4.  But  the  distance 
(apparent)  from  center  to  center,  we  have  just  determined  to  be 
2106"7  ;  therefore  the  distance  from  limb  to  limb  must  be  255"3, 
and  the  eclipse  has  not  yet  commenced,  and  cannot  commence, 
until  the  moon  gains  256"  on  the  sun's  motion,  which  will  require 
more  .than  ten  minutes  of  time. 

We  now  require  the  apparent 
distance  between  the  centers  of 
the  sun  and  moon,  ten  or  twelve 
minutes  later,  so  as  to  get  the  ap- 
parent rate  of  approach.  The  rate  ^^^ 
is  continually  changing,  but  du-  ^Qf^ 
ring  any  short  interval  of  ten  or 
twelve  minutes,  it  may  be  consid- 
ered uniform,  without  any  sensible  error. 

If  we  vary  the  time,  the  angle  ZPS  will  vary  1°  to  4  minutes, 
but  in  that  variable  time  the  moon  will  move  from  c  to  m,  and 
the  angle  ZPm  will  vary,  but  not  quite  so  much  as  ZPS.  The 
question  now  is.  If  we  make  a  small  difference  in  the  angle  ZPm, 
what  corresponding  difference  will  it  make  in  the  arc  Zm  ;  and  this 
is  a  question  in  the  differential  calculus,'^  although  we  can  work 
it  out  at  large  by  spherical  trigonometry. 

We  will  take  the  interval  of  12m.,  then  the  angle  ZPS  will 
increase  3°.  But  in  one  hour  the  moon's  motion  in  right  ascen- 
sion exceeds, that  of  the  sun  28'  47"  ;  this  in  12m.  will  be  5'  45"4, 
therefore  the  angle  ZPm  varies  in  that  interval  of  time,  2°  64' 
14"6 =2.90406,  taking  one  degree  as  the  unit. 

*  The  differential  calculus  is  the  science  of  minute  variations,  or  of  corres- 
ponding small  differences — a  science  which  owes  its  birth  to  the  varying 
dfeements  of  astronomy. 


ASTRONOMY.  30S 

The  equation  as  before  is 

sin.  ^=cos.  Zw=cos.  P  cos.  L  sin.  D-\-m!L,  L  cos.  D 
But  we  have  caused  P  to  vary,  while  L  and  D  remain  constant. 
What  variation  will  this  give  to  the  altitude  A  ? 
Taking  the  differential  of  the  equation,  we  find 

*  jM sin.  P  COS.  L  sin.  B-dP 

cos.  ^ 
But  we  have  assumed  c?P=2. 90405,  while  P,  Z,  B,  and  -4,  in 
this  equation,  have  the  same  values  as  before.      That  is, 

/>=61°  30'  45",  i;=44°  16'  33",  i>=68°  26'  28",  and  ^=34° 
58'  10". 

log.  2.90405 0.463000 

sin.P —1.943954     (radius  unity.) 

COS.  L — 1.854910 

sin.  B —1.968853 

COS.  complement  A 0.086445 

rf^=2.0755 0.317162 

Thus  we  find  that  the  moon  changes  its  altitude  at  this  time, 
in  the  interval  of  12  minutes,  .2°  4'  32",  and  because  the  second 
member  of  the  last  equation  is  minus,  the  altitude  has  diminished. 

Moon's  altitude  was 34°  58'  10" 

Variation 2°    4' 32" 

Moon's  altitude  at  this  time 32°  53'  38" 

The  angle  ZPm=61°  30'  45"+2°  54'  15"=64°  25'. 
cos.  32°  53'  38"  :  sin.  64°  25'  :  :  cos.  44°  16'  33"  :  sin.  ZmP=sin. 
60°  16'  30". 

To  find  mn,  or  the  parallax  in  altitude. 

To  log.  of  the  horizontal  parallax , 3.513044 

Add  COS.  32°  53'  38" 9.924100 

Approximate  value  of  mn  45'  36"=2736" 3.437144 

From  the  moon's  true  alt.  32°  53'  38" 

Subtract  apparent  parallax        45'  36"  3.513044 

Moon's  appa.  alt.  nearly   32°    8'    8"    cos 9.927877 

True  value  of  mn  2760" 3.440921 

As  before, 

E     :     2760     :    :     sin.  50°  16'  30"     :    p 
R     :     2760     :    :     cos.  60°  16'  30"     :     q 


4 

?i04  ROBmSON'S  SEQUEL. 

3.440921  3.440921 

sin.  50°  1 6'  30" ..9.885996  cos 9.805420 

J^2123" 3.326917  q   1763"5.. .  .3.246341 

During  the  12  minutes  the  moon  moves  over  the  oblique  stkuiU. 
arc  C7n,  (in  relation  to  the  sun,  as  conceived  to  be  stationary,) 
which  is  5'  45"4  in  right  ascension,  or  the  difference  between  the 
two  meridians  PS  and  Pm  on  the  equator,  is  5'  45"4,  or  345"4. 
The  perpendicular  distance  at  the  point  m  is  therefore  found 
by  multiplying  345"4  by  the  cosine  of  the  moon's  declination  to 
radius  unity.     Therefore, 

Log.  345"4 2.538322 

Moon's  dec.  21°  35'  nearly,  cos 9.968429 

Perpendicular  dis.  between  PS  and  Pm  32r'2.  ..2.506751 
During  one  hour  the  moon's  relative  motion  in  declinatian  is 
7'  41"4.  During  12  minutes  it  is  therefore  92"2,  which  added  to 
22' 15"  or  1335"  makes  1427"2  for  the  distance  represented  by 
ma.  But  q  1763"5  is  the  effect  of  parallax  on  the  line  or  the  ef- 
fect in  declination,  and  it  being  greater  than  1427"2,  their  differ- 
ence, 336"2  is  the  apparent  distance  in  declination  of  n  below  S, 
or  of  the  center  of  the  moon  below  the  center  of  the  sun. 

Again,  p  is  the  parallax  in  right  ascension,  projecting  the  moon 
2123"  west  of  its  true  place,  while  it  is  321  "2  east  of  the  sun  ; 
therefore  the  apparent  right  ascension  of  the  moon  is  1801  "8  west 
of  the  sun.  Consequently  the  apparent  distance  of  the  two 
centers  is 

V(1801"8p+(336"2)2  =  1 833"2. 
But  the  semidiameter  of  the  sun  and  the  augmented  semidiam- 
eter  of  the  moon  at  this  time  amount  to  1850"4,  differing  only 
17"2.  The  distance  between  the  centers  being  less  than  the  sum 
of  the  semidiameters,  shows  that  the  eclipse  has  already  com- 
menced 

Twelve  minutes  before  this  time,  the  distance  between  the 

centers  was ■* 2105"7 

Now  it  is 1833"2 

Moon's  apparent  motion  in  12  minutes . . .'. 272"5 

or  22^7  in  one  minute. 

Then        22"7     :     17"2     :    :     60s.     :     46.4  seconds. 


ASTRONOMY.  306 

That  is,  the  eclipse  commences  11m.  14.6  sec.  after  the  appa- 
rent time  of  conjunction  at   Burlington,  or  at  4h.  17m.  17.6  sec. 

If  6  seconds  be  taken  from  the  sum  of  the  semidiameters  for 
irradiation  and  inflection,  as  most  astronomers  recommend,  the 
eclipse  will  commence  at  4h.  17m.  30s. 

THE    POINT    OF    FIRST    CONTACT. 

The  point  n,  the  apparent  place  of  the  center  of  the  moon  is 
nearly  west  of  S  and  the  angle  ZmP=50°.  Therefore  the  point 
of  first  contact,  from  the  sun's  vertex,  must  be  (50"-}~^0°)»  ^40° 
towards  the  right,  but  if  viewed  through  an  inverting  telescope, 
the  appearance  will  be  directly  opposite. 

GREATEST    OBSCURATION. 

The  time  of  greatest  obscuration  will  take  place  not  far  from 
Ih.  20m.  after  conjunction  at  Burlington,  or  not  far  from  5h.  26m. 
3s. ;  we  will  therefore  compute  the  apparent  distance  between  the 
two  centers  for  this  time.  We  could  compute  it  by  proportion, 
provided  the  apparent  motion  of  th,e  moon  was  uniform,  and  in  a 
straight  hne  ;  but  that  motion  being  neither  uniform  nor  in  a 
straight  line,  we  are  compelled  to  compute  it  by  points  to  obtain 
any  thing  like  accuracy. 

Using  the  last  figure,  the  angle  ZP^=5h.  26m.  3s. =81^  30' 
45"  ;  but  during  Ih.  20m.  the  moon  will  gain  38'  22"  in  right  as- 
cension ;  therefore  the  angle  ZPw=80°  62'  23". 

In  Ih.  20m.  the  moon  will  increase  her  declination   10'  49", 
making  it  21°  44'  21",  or  Pm=68°  15'  39",  and  am  is  now  32' 
30"=  1950". 
As  before, 
sin.  -4=cos.  Z7»=cos.  P  cos.  L  sin.  i)-|-sin.  L  cos.  D. 

COS.  P=cos.  80°  52'  23" 9.200404 

cos.  Z=cos.  44°  16'  33" 9.854910     sin 9.843917 

sin.  i>=sin.  68°  1 6'  39" .9.967959     cos 9.568656 

0.10555. —1.023273  .25856  —1.412571 

Nat.  sin.  ^=0. 10555+0.25856=.3641 1 . 

Whence,  A,  moon's  true  alt.=21°  21'  12".     Zm=68°  38'  48". 
20  ' 


% 


306  ROBINSON'S  SEQUEL. 

sin.  68°  38'  48"  :  sin.  80°  52'  23"   ;  :  cos.  44°  16'  33"  :  sin.  ZwP= 
sin.  49°  22'  45". 

To  find  mn.     Moon's  horizontal  par.  log 3.513044 

COS.  21°  21'  12" 9.969114 

Approximate  value  of  mn  50'  35"=3035" 3.482168 

From  moon's  true  alt. .  .21°  21'  12" 

Take 50'  35"  3.513044 

Moon's  appa.  alt.  nearly  20°  30'  37"        cos 9.970630 

True  value  of  mn  3046" 3.483674 

R     :     3045"     :    :     sin.  49°  22'  45"     :    p 
B     :     3045"     :    :     cos.  49°  22'  45"     :     q 

3.483674  3.483674 

sin.  49°  22'  45" 9.880265         cos 9.813620 

i?=2311"8 3.363939     ^=1982"6...  .3.297294 

In  the  Ih.  and  20m.  which  elapses  after  conjunction,  the  moon 
gains  38'  22"  or  2302"  in  right  ascension  on  the  sun  ;  but  this  is 
arc  on  the  equator,  it  is  not  perpendicular  distance,  the  two  me- 
ridians PS  and  Pm,  drawn  from  m ;  but  that  distance  is  required 
and  it  is  found  thus  : 

Log.  2302 3.362105 

Add  COS.  of  moon's  declination  21°  44'  21" .9.967959 

wic 2138"5 3.330064 

p 231 1"8 

Moon  apparently  west . .  173"3 

Moon's  declination  north  of  sun  am 1960" 

Moon's  parallax  in  declination  q 1982"6 

Moon  apparently  south  of  the  sun 32"6 

Distance  between  centers= ^(iTS^p  +(32"6) ^  =  1 76"3. 
We  know  by  comparing  this  result  with  the  last,  that  the  gi-eatest 
obscuration  or  nearest  approach  of  the  centers,  must  take  place 
about  7  minutes  after  this  time.      We  will,  therefore,  differentiate 
for  10  minutes. 

In  10  minutes  the  sun's  polar  angle  will  increase  from  the 
meridian  2°  30' 

For  the  3'^  motion  in  R.  A.  sub.         4'  47" 
The  angle  ZPm  will  increase      2°  26'  13  '=2°.4202. 


ASTRONOMY.  307 

As  before, 

.  _     sin.  P  COS.  L  sin.  D  (  g.4202) 
COS.  -4 
An  equation  in  which  ^=21°  21'  12",  P=  80°  62'  23",  Z= 
44°  16'  33",  and  i>=68°  16'  39". 

sin.  P —1.994465  (radius  1) 

COS.  L —1.864910 

sin.  i> —1.967959 

log.  2.4202 , 0.383861 

COS.  complement  A 0.030886 

(^-4=1.7062 0.232071 

The  minus  sign  before  the  second  member  shows  that  this  must 

be  subtracted  from  A.  A 21°  21'  12" 

1.7062=  1°  42'  22" 

Moon's  true  altitude  at  this  time, 19°  38'  60" 

Log.  Horizontal  parallax 3.513044 

cos.  19°  38'  60" 9.973950 

51'    9" 3.486994 

Moon's  app.  alt.  nearly  18°  47'  41" 

3.613044 

cos.  18°  47'  41" 9.976154 

True  value  of  mn         3084"5 3.489198 

cos.  19°  38' 60"  :  sin.  83°  17' 36"  :   :   cos.  44°  16' 33"  :  mi.ZmP 
=sin.  49°  1'  40". 

R     :     3084"6    :    :     sin.  49°  1'  40"    :    ^ 

R    :    3084"6     ::     cos.  49°  1' 40"    :    g 

3.489198  3.489198 

8in.49°  1^  40"  9.877978         cos .9.816700 

p  2329". . . .  3.367176     g  2022"7 3.306898 

At  the  last  point,  am  was  1950"  which  has  increased  77"  by  the 
moon's  motion  ;  therefore  it  is  now  2027".  - 

At  the  last  point,  Sa  was  2302"  of  arc  which  has  increased 
287",  making  2589",  which  must  be  reduced  to  the  arc  of  a  great 
circle  as  before. 

f 


!► 

^ 


ROBINSON'S  SEQUEL. 

Log.  2589 3.413132 

Moon's  declination  21°  46'         cos 9.967927 


Moon  east  of  sun 2404"6 3.381069 

Parallax  in  R.  A.    p.,  .2329" 
Moon  east  of  sun,  apparently    76"6 

Moon  north  of  sun 2027" 

Parallax  in  declination,  q., ,,  2022"7 
Moon  north  of  sun,  apparently ....      4"3 


Distance  between  centers  =  ^(76"6)2-|- (4"3)=75''6,  appa- 
rently. 

Now  to  find  the  nearest  approach  of  the  centers,  the  time  of 
forming  the  ring,  its  continuance,  &c.,  we  have  a  very  delicate 
and  simple  problem  in  plane  geometry. 

Let  S  be  the  center  of 
the  sun.  Take  SV  = 
173"3,  F"u4=:32"6.  Then 
^aS'=176"3.     Also  take 

then  /Sf^=75"6,  BD= 
173"3+75"5=248"8.      i>^ =37"  nearly  ;  then  ^J5  the  moon's 
apparent  motion  on  the  face  of  the  sun  during  10  minutes  must  be 
V(248"8)2-f(3rp  =251  "5. 
Therefore  the  apparent  motion  per  minute  is  25"15. 
We  must  now  find  Sm,  the  distance  between  the  two  centers  at 
the  time  of  their  nearest  approach.      In  the  triangle  ABS,  we 
have  all  the  sides,  therefore  by  (Prop.  6,  page  149,  Geom.),  we 
have      AB     :    AS-\-SB     :    :     AS—SB     :    Am-^B 
That  is,  26r'6  :  26r'9    :    :    100.7  :  100.86 
^wi+7w^=251.6 
Am—mB=100M 

2mJ5=  150.64         m.B= 76.32. 
Whence,  Am=n&'lB.     In  the  right  angled  triangle  BmS,  we 
have 

Sm=  J(75"6y—{75"32y  =6"5 

Moon's  semidiameter  from  giv^a  elements 14'  63"6 

Augmentation  for  alt.  18°  <""  (see  table) 4^7 


ASTRONOMY.  309 

Augmented  semidiameter 14'  58"2 

Sun's  semidiameter  from  given  elements 15'  48"9 

Diflference, 50"7  * 

It  is  obvious  that  the  ring  will  form  when  the  distance  between 
the  two  centers  comes  within  the  difference  of  the  semidiameters. 
Suppose  it  to  form  when  the  moon's  center  passes  n  ;  then  in  the 
right  angled  triangle  Smn,  Sn=50"7.     Sm—6"5. 
And  mw=7(50"7)2— (6"6)2=50"28. 

An=Am — mw=126"9,  and  at  the  rate  of  25"16  per  minute, 
this  will  be  passed  over  in  5m.  2.7  seconds,  nm  in  2  minutes  very 
nearly,  and  an  equal  line  on  the  other  side  of  m  in  2  minutes 
more. 
The  appa.  time  the  Q)'s  center  arrives  at  A  is  5h.  26m.     3s. 

To  which  add 5m.  2.7s. 

Ring  forms  at 5h.  31m.  5.7s. 

Time  of  nearest  approach 5h.  33m.  5.7s. 

Rupture  of  the  ring 5h.  35m.  5.3s. 

At  the  time  of  nearest  approach  the  breadth  of  the  ring  on  the 
north  limb  of  the  sun  will  be  ST'O,  and  on  the  south  limb  18"8  ; 
but  if  the  customary  allowance  be  made  for  irradiation  and  inflec- 
tion, these  quantities  reduce  to  31  "3  and  18"3,  and  the  duration 
of  the  ring  must  be  reduced  from  3m.  59.6s.  to  3m.  55.2s. 

THE    END    OF    THE    ECLIPSE. 

We  know  by  the  moon's  apparent  motion  (25''15  per  minute, 
which  is  continually  increasing)  that  more  than  an  hour  will  be 
required  from  the  time  of  nearest  approach,  for  the  eclipse  to  pass 
ofif.  We  will  therefore  compute  the  apparent  distance  between 
the  two  centers,  one  hour  and  ten  minutes  after  the  moon  passes  A^ 
of  the  last  figure. 

By  referring  back  we  shall  find  that  the  point  A  corresponds 
with  6h.  26m.  3s.  or  81°  30'  45"  for  the  angle  ZP5.  One  hour 
and.  ten  minutes  later  will  be  6h.  36m.  3s.  and  will  correspond  to 
99°  0'  45"  for  the  angle  ZPS.     (See  next  figure.) 

*  Astronomers  recommend  a  diminution  of  3"  for  the  sun's  semidiameter 
for  irradiation,  and  a  diminution  of  2"  of  the  moon's  semidiameter  for  in/lio^ 
Hon,  this  would  make  49"7  for  their  difference  instead  of  50"7. 


•  ' 


310  ROBINSON'S  SEQUEL. 

Butthe  difference  between  the  right  ascensions  of  the  sun  and 
moon  is  now  1°  21'  66"  ;  therefore  the  angle  ZFm=97°  38'  49". 
At  5h.  26m.  3s.  the  value  of  ma  was  1950",  in  one  hour  and  ten 
minutes  it  increased  540",  it  is  now  2490". 

Sa  in  arc=l°  21'  66"= 49 16"  which  we  reduce  to  distance. 

Log.  4916 3.691612 

Sun's  dec.  cos.  21°  12' 9.969667 

Sa 4583" 3.661179 

As  before, 

sin.  ^=cos.  Zm=cos.  P  cos.  L  sin.  D-j-sin.  L  cos.  D 
cos.  P=cos.  97°  38'  49".  .—1.124088 
cos.  L  =cos.  44°  16'  33",  .—1.854910     sin..  .—1.843917 
sin.  D  =sin.  68°    7'  13".  .—1.967531     cos.  .—1.671327 

—0.088416* —2.946529  0.26015-1.415244 

sin.  A  or  cos.  Zm=0.26015 — 0.08842=. 17173 
Whence,  ^=9°  53'  21".     Zw=80°  6'  39". 
sin.  80°  6' 39"  :  sin.  97°  38' 49"  :  :  cos.  44°  16' 33"  :  sin.Zwi'= 
sin.  46°  6'. 

Log.  Horizontal  parallax 3.613044 

cos.  9°  63'  21" 9.993500 

63'  30" 3.506544 


Moon's  app.  alt.  8°  69'  61"     cos 9.994618 

3.613044 

mn 3.607662 

sin.  46°  5'     :    p 
cos.  46°  6'     :     g 

3.607662 

cos 9.841116 

p  2918.4 3.465192     q  2232.4 3.348778 

Sa  4583.  2490 

~1664.6  267.6 


H    : 

3218     : 

B    : 

3218     : 

3.507662 

'5'... 

..9.857530 

Distance  between  the  centers  =  7(1 664.6 )2-f-(267.6)»  =  1674"5. 

*  This  number  must  be  minus  because  cos.  97°^  is  minus,  the  cosine  of  any 
arc  over  90°,  as  far  as  270°,  is  minus. 


m 


ASTROHOMY.  Sll 

The  sum  of  the  semi- 
diameters  is  now  1 843" ; 
therefore  the  sun  is  still 
eclipsed,  and  will  be  un- 
til the  apparent  motion  of 

the    moon    passes    over    ^V^^^^^^^^HB^ 
169",  which  will  require 
a  little  over  6  minutes  of 

time,  we  will  therefore  compute  the  apparent  distance  of  the  cen- 
ters for  8  minutes  later.  In  8m.  the  angle  ZPS  will  increase 
2°,  and  the  angle  ZFm  will  increase  2°— (3'+50")  or  1^  66'  10". 
At  the  last  operation  ZFm  was  97°  38'  49",  now  it  must  be  99° 
34'  59",  say  99°  35'. 

If  we  take  D  as  constant  in  the  last  equation,  we  shall  find 
that  all  our  logarithms  will  be  the  same  except  cos.  F,  and  all  we 
have  to  do  is  to  add  to  log.  — 2.946529  the  difference  between 
COS.  F  in  the  last  operation  and  the  cosine  of  99°  35',  or  the  sine  of 
9°  35'  for  the  log.  of  the  first  number  composing  the  natural  sine 
of^. 

That  is,  to —2.946529 

Add .097279        ' 

Number,  —0.110604 —1.043808* 

sin.  u4=0.26015— 0.1 10604=. 149546.  Whence,  ^=8°  36'.t 
COS.  8°  36'  :  sin.  99°  35'  :  :  cos.  44°  16' 33"  :  sin.  ZwP=sin. 
45°  33'  50". 

Log.  Horizontal  parallax 3.513044 

COS.  ^  8°  36' 9.995089 

Approx,  val.  of  mn      53'  42" 3.508133 

Q)'s  appa.  alt.  nearly  7°  42'  18"    cos 9.996064 

3.513044 

True  value  of  rm  3229"5 3.509108 

JR     :     39.2d"5     :    :     sin.  45°  33'  50"     :    p 
R     :     3229"5     :    :     cos.  45°  33' 50"     :     q 

♦Artifice  should  be  employed  to  take  out  the  number  corresponding  to  this 
logarithm,  such  as  is  taught  in  the  author's  Surveying  and  Navigation. 

t  Here  we  have  jumped  from  one  result  to  another,  and  did  not  obtain 
the  difference  between  one  result  and  another,  as  we  do  by  the  differential 
method. 


312  ROBINSON'S  SEQUEL. 

3.509108  3.509108 

sin.  46°  33'  50\  ...9.853717     cos 9.845168 

^  2902"6 3.462825     q  2260.6 3.364276 

Before  Sa  was  in  arc  4916",  increase  in  8m.  230";  therefore  it 
is  now  6146"  which  must  be  reduced  as  before. 

Log.  6146. 3.711470 

cos.  21°  12'. .  ...9.969667 
Sa  in  space, . .  .4797"6. .  .3.681037 

p.... .2902"6  ma  before  was 2490 

Paral.  in  R.  A — 1896  Licrease  in  8m 62 

2662 

q 2260.6 

Moon  apparently  north  of  sun 291 .6 


Distance  between  the  centers=^(1895)2-|-(291.5)2=  1917"3. 
This  being  greater  than  1843"  shows  that  the  eclipse  has  passed 

oior. 

The  distance  between  the  centers  now  is 191 7"3 

Eight  minutes  ago  the  distance  was 167^45 

Apparent  motion  of  the  moon  in  8  minutes 242"8 

Corresponding  motion  for  1  minute  30"35. 
From  the  sum  of  the  semidiameters  1843",  subtract  1674"5,  and 
we  obtain  168"5  for  the  moon  to  pass  over  before  the  end  of  the 
eclipse.     This  at  the  rate  of  30"36  per  minute  requires  5m.  33s. 

Hence,  to 6h.  36m.    3s.  appa.  time. 

Add 5m.  33s. 

Eclipse  ends 6h.  41m.  36s.  appa.  time. 

But  if  we  reduce  the  semidiameters  for  irradiation  and  inflection 
6",  then  we  must  diminish  the  time  of  ending  10  seconds. 

We  may  now  observe  that  the  moon's  apparent  motion  across 
the  sun  was  at  the  rate  of  22"7  per  minute  at  the  beginning  of 
the  eclipse,  25"15  at  the  time  of  nearest  approach,  and  30"35  per 
minute  at  the  end.  This  variability  of  the  apparent  motion  is 
owing  to  the  varying  effects  of  the  moon's  parallax  correspond- 
ing to  the  different  altitudes,  and  this  makes  the  problem  tedious, 
and  throws  over  it  an  air  of  complexity. 
Since  six  o'clock  the  rate  of  the  moon's  motion  from  the  sun 


ASTRONOMY.  ^» 

increased  very  much,  and  any  one  can  see  the  rationale  of  this  by 
inspecting  the  projection  on  page  293  of  Robinson's  Astronomy. 

Along  the  mid- day  hours  the  sun  and  moon  have  an  apparent 
motion  together,  but  with  diflPerent  velocities.  As  the  time  from 
noon  increases,  the  sun's  motion  along  the  ellipse  is  slower,  and 
the  moon  appears  to  run  over  it  faster  and  faster.  After  6  the 
sun's  apparent  motion  is  no  longer  with  the  moon's,  hence  a  rapid 
increase  in  the  moon's  apparent  motion. 

We  have  made  the  problem  much  longer  than  we  should  have 
done,  had  we  simply  been  in  pursuit  of  results.  Our  object  has 
been  to  explain  and  illustrate  the  problem  to  a  learner,  through 
each  consecutive  step,  and  we  have  found  the  following 

SUMMARY. 

Appa.  time  Burlington,  Vt.  Mean  time. 

^     Beginning  of  the  eclipse,         4h.  17m.  17.6s.      4h.  14m.    2.5b. 

Formation  of  the  ring,  5h.  31m.    5.7s.      5h.  27m.  50.6s. 

Time  of  nearest  appr.  of  cen.  5h.  33m.    5.7s.      5h.  29m.  50.6s. 

Rupture  of  the  ring,  5h.  35m.    5.3s.      5h.  31m.  50.2s. 

End  of  the  eclipse,  6h.  41m.  36    s.      6h.  38m.  21    s. 

Duration  of  the  ring  4  minutes  nearly ;  duration  of  the  eclipse 

2h.  24m.  19s.     When  the  ring  is  most  perfect,  its  breadth  on  the 

north  limb  will  be  31",  and  on  the  south  limb  18". 


Not  long  since  the  author  received  the  following  request :  we 
extract  from  the  letter. 

**  One  request  more.  In  your  Astronomy,  page  191,  near  the 
bottom,  you  say,  (speaking  of  the  radial  force,)  'and  the  diminu- 
tion in  the  one  case  is  double  the  amount  of  increase  in  the  other,  and 
by  the  application  of  the  differential  calculus,  we  learn  the  mean 
result  for  the  entire  revolution,  is  a  diminution  whose  analytical 

rS 
expression  is  ;  an  expression  which  holds  a  very  prominent 

place  in  the  lunar  theory.* 

r  Sf 

Now  my  enquiry  is,  how  can  we  obtain  the  expression for 


314  ROBINSON'S  SEQUEL. 

the  mean  result  ?  What  operation  in  the  calculus  shall  we  go 
through  ? 

Yours,  &c.,  Wm.  T  " 

To  this  we  returned  the  following  reply  : 

On  page  193  you  will  find  the  following  expression, 

4^  rS (  S COS.' X— I) 

for  the  radial  force  corresponding  to  any  angle  x  from  the 
syzigies. 

We  already  know  the  value  of  this  force  at  the  syzigies  and 
quadratures,  and  at  these  points  the  result  has  the  same  general 
form ;  therefore  the  result  for  the  entire  quadrant,  that  is,  the 
mean  result  for  the  whole  quadrant,  will  be  found  by  taking  the  an- 
gle ar=45°,  and  as  the  mean  result  for  each  quadrant  is  the  same, 
this  will  be  the  mean  result  for  the  entire  revolution. 

Whence,  a;=46°,  sin. a;=cosic,  and  2cos.^a;=l,  or  cos.^a:=|. 

Or,  (Scos.^ar — 1)=^  ;  whence  the  above  general  expression 

becomes  ~-. 
2a3 

To  this  was  returned  the  following  observation : 

*'I  understand  your  explanation,  it  is  very  simple  ;  but  why 
did  you  not  make  this  explanation  in  the  book, — and  more  than  all, 
why  do  you  call  it  an  application  of  the  differential  calculus  ?  /can 
see  no  calculus  in  it. 

Yours,  &c.,  Wm.  T." 

To  this  we  rejoined  as  follows  : 

If  the  operation  I  sent  you  is  not  calculation,  I  know  not  what 
it  is — it  may  therefore  be  called  calculus  ;  ahd  if  in  any  operation 
small  quantities  may  be  omitted  on  account  of  their  insignificance 
in  relation  to  larger  quantities,  the  small  difference  so  omitted 
constitutes  the  differential  calculus,  and  to  obtain  that  geneial 
expression,  you  will  see,  by  looking  on  page  193  of  the  Astronomy, 
that  the  powers  of  r  above  the  first  were  omitted. 


CALCULUS.  316 

THE   €AjL€UI.irS. 

DIFFERENTIAL    CALCULUS. 

The  differential  calculus  is  a  branch  of  Analytical  Geometry. 
It  k  a  science  for  computing  the  ratio  of  small  diflferences. 

For  example,  the  side  of  a  square  is  increased  by  a  very  small 
quantity,  what  will  be  the  corresponding  increase  of  the  square 
itself  ? 

The  side  of  a  cube  is  increased  or  diminished  by  a  quantity 
very  small  in  relation  to  the  side  itself, — ^how  much  will  this  in- 
crease or  diminish  the  cube  ? 

The  arc  of  a  circle  is  increased  or  diminished  by  a  quantity  very 
small  in  comparison  with  itself,  what  effect  will  this  have  on  the 
sine  and  cosine  of  the  arc  ? 

The  sun's  longitude  increases  a  certain  distance  in  10  minutes, 
what  is  its  corresponding  change  in  declination  ?  Or,  find  the 
law  of  these  corresponding  changes,  or  differences — called  differ- 
etUicds,     These  questions  explain  in  part  the  object  of  the  calculus. 

The  calls  of  astronomy  gave  birth  to  this  science,  as  we  have 
before  remarked. 

For  the  development  of  this  science,  see  the  various  works  upon 
it.  We  confine  ourselves  in  this  book  to  a  few  difficult  or  curious 
operations. 

We  presume  the  reader  is  acquainted  with  all  the  rules  of 
operation. 

EXAMFLSS. 

(1.)     Differentiate  the  expression  jY—x^.    Ans. 


Jl—x' 


Put    u=Jl—x^.     Square,  u^=:l—x^  ;  then 

xdx 


2udu= — 2xdXf  or  du^=-d*  J\ — x^=- 


Jl—x' 

(2.)     Find  the  differential  of  the  equation 

X 

u= 


516  ROBINSON'S  SEQUEL. 

By  the  rule  for  diflferentiating  a  fraction,  we  have 

x'dx 


dx(x-\-  J 1  — x^) — xdx- 


{x+j\-x-y 


J\-X-+-^^:^-_ 

dx        (^x+JU^"")^ 
By  multiplying   numerator   and   denominator  of   the    second 


member  by  ^1 — x^  y  then  multiplying  the  equation  by  dx,  we 
have 

J  dx 

du= 


{x+J\-^x^Y  J\—x' 


(3.)     Q\yenu^=la-\-Jb — —\    to  find  the  differential  of  u. 

Put  y=J^^ — —  and  extract  the  4th  root  of  the  original  equa- 
tion ;  then 

I  tt*~  du=dy 

du:=^dy{a-\-yy  (1) 

cdx 


But    y^  =  h — ~    Whence,  yc?y= — _.  dy 


2  *-        x^       "       ^Ah-^— 


Substituting  the  values  of  y  and  £?y  in  (1),  we  have 


v- 


CALCULUS.  317 


/I  \  X  \  J\ X 

C4.)    Given  u=~  ■      ^  to  find  the  differential  of  u* 

^    ^  Jl-\-x—Jl—x 

Reduce  the  second  member  by  multiplying  numerator  and  de- 
nominator of  the  fraction  by  the  numerator ;  then 


X 

Apply  the  rule  to  differentiate  a  fraction,  and  to  differentiate  the 
numerator,  simply  make  use  of  the  first  example. 

—  x^dx 


du 


J^Z^-d<^+J^-^') 


X' 


Dividing  both  members  by  dx,  and  changing  signs,  and 

~dx     Jl-x-'^         X-  ^^^ 


X^J\—X^  X^'Jl- 


n-\-J\—x^\ 

Whence,  du= — I )dx 

\  x^J\—<c^  / 


(l\  \  X  \  J\ x\ 
^~=---'^^=_-  )  to  find  the  differential 
J\J^x—J\—x/ 

of  u.\ 

The  differential  of  a  logarithm  is  the  differential  of  the  quan- 
tity divided  by  the  quantity.  But  we  have  just  obtained  the 
differential  of  the  quantity  in  the  4th  example,  therefore  all  we 
have  to  do  is  to  divide  that  result  by  the  quantity  itself. 

/1_L  /l_a:2v  X  dx^_ 

*  These  examples  are  common  to  all  or  nearly  all  the  works  on  the  calculus ; 
they  are  in  the  works  of  La  Croix,  from  which  they  have  been  extracted  into 
other  works. 

+  This  example  is  worked  differently  in  Davied'  Calculus,  page  63 ;  the 
work  is  extracted  from  La  Croix. 


S18  ROBINSON'S   SEQUEL. 


(6.)     Given  «=  log.  (ar+^l-f-a;')  to  find  the  diflferential  of  u. 
By  the  rule  for  differentiating  a  logarithm,  we  have 


du=         Jl+x' 


x+Jl+x' 
To  simplify  the  operation,  divide  both  members  by  dx,  then 

II  X 

du     H 
dx 


-=.     JXJ^ 


x^J\-\-x- 
Multiply  numerator  and  denominator  of  the  second  member 

by  V  H-^  then 


Whence,  c?«= 


dm ^i-far^+a; ^ 

dx 


(7.)     Given  «=_?_  log.  {xJ^^\-\-J\^l^)   to  find  the 

differential  of  u. 

To  avoid  the  confusion  of  mind  which  naturally  accompanies 
the  imaginary  symbol  J — 1,  I  put  a=^ — 1.     Then, 


att=log.  {aX'\-J\-\-x^) 

J           xdx 
adx — 


aX'\-Jl — x'* 

dx       . V-hlf-L     (a»+Vl— «')n/1— «' 

ax-\'Jl — x^ 

Now  divide  the  numerator  in  the  second  member  by  the  first 
factor  in  the  denominator,  thus  : 


Jl—^^'+ax  )  aj\—x^—x  (  « 
aJ~U^x^+a^x 


CALCULUS.  3je 

There  is  no  remainder  because  a^a:= — Xy  as  will  be  obvious  wben 
we  consider  a=J — 1  ;  hence  a^= — 1. 

Whence,         <^^^^ — ^— .  or  du=:     ^ 

dx      J\—x^  Jl^x^ 

(8.)  Given  w=log.  (V^+^HlfV    to  find  the  diflferential 

Reduce  the  fraction  by  multiplying  numerator  and  denominator 
by  the  numerator,  then 


«=log.(! 


That  is,  «=log.  (x-\-Jl-\-x'),  and  this  is  the  6th  example. 
Therefore,  du=      ^    . 

(9.)     Differentiate  the  equation  w=a;'"(log.  a?)". 
Put  log.  x=z  ;  then  u=x^z'^ . 
And  du=mx'^^z''dx-\-nz'^'^x^dz. 

Because  2=log.  x,  dz= —    Now  by  substitution, 

X 

du=mx"^^  (log.  xydx-^n(\og.xy-^a^^dx 

Finally,    ^=(mlog:a;+»>°*-i(log.a;)°->. 
dx 

(10.)     mffereniiateu=^'^^"^^^Mf)+'^ 

As  before,  let  0=log.a; ;  then  the  equation  becomes 
_j:^z^ x*ZjX^ 

^       4~    ~8"^32 
Then  ^./  =  ^^,,2^^xx'zdz_x^zdx_x^ dz.x^dx 

'2  2  8^8 

But  dz= — ,  and  z  =  log.a; ;  substituting  these  values,  the  pre- 

X 

ceding  equation  becomes 

du=xH\09r  ^y^^ji^''(^^g'^)dx_x^(\og.x)dx___x^dx,x^dx 
V    e-  y       -r         2  ^  8^8 


Whence,  du=x^{\og.xydx. 


320  ROBINSON'S  SEQUEL. 

(11.)  Differentiate  u=\og.^x  :  that  is  to  say,  the  logarithm  of 
the  logarithm  of  x. 

Put  log.  x=:z.     Then  w=log.  z. 

And   du=—.     But  dz=^. 

z  as 

That  is,  du=—= -— 

zx    a;(log.a:) 

(12.)     Differentiate  u=\og.^x.  '^ 

By  the  aid  of  the  previous  example  we  learn  that 

log.5a;=log.(log.''a;). 

Whence,         du='iJ^?L^  (1)         . 

log.  4:?; 

d  (  log.*a;)=-i — 5.' — L,  this  substituted  in  (1)  reduces  it  to 
log.^a; 

du=       ^(l^g-^^) 
(log.'»a;)(log.3^) 

Another  step  gives    du= _i— ^l_^i 

(I0g.^..)(l0g.=^2r)(l0g^a:) 

Using  the  result  of  the  previous  example,  we  finally  have 

,  dx 

du= 

(log.'*a:)(log.^a;)(log.^a;)(log.a;)a; 

(13.)     Differentiate    u=e  (x —  1 ) . 

Here  we  must  take  the  logarithm  of  each  member,  observing 
that  the  log.  of  e^  is  simply  x,  because  e  is  the  base  of  the  Na- 
perian  system  of  logarithms,  the  system  always  used  in  such 
examples. 

log.  w=a;-|-log.(a: — 1). 

du      ,    ,    dx         dx 

— =dx-\- = 

u  X — 1     x — 1 


u 


_e\x-\). 


Whence,  r:!!=_Jl_=lAlIZiZ=c\     Or,  du^edx, 
dx    X — 1        X — 1 

,(14.)     Differentiate  u=^e'{x'^—^x'^'\-^x—Q). 
log.w=^-|-log.(4;^ — 'ix^-\-^x — 6) 
du_.    .Sx^dx — Gxdx-\-6dx 
w'""  x^-^x^'+ex—G 

du x^ 

udx^'x^-^Sx^-^-Qx^ 


CALCULUS.  321 

du  x^u 


dx     x^'—^x'^'+Qx—Q 
(15.)     Differentiate    u= 


Or,  du^e^x^dx. 


\—x 

u(\ — x)^=e'x.     log.w-|-log.(l — x)-=x-\-\q^.x. 

dxL       dx  7    [  dx  du 1      .  x-\-\ \-\-x — x^ 

u      1 — X  X  udx     1 — X        X         (1 — x)x 

du_{\^x—x^)       e^x  _[\J^x—x'')e^  ^^  Jw=(i+fr:^!if!^ 
dx        {\—x)x        \  —  x     '     {\—xY  '  (1—^)^ 

(16.)     Dffererdiate     uz^e'log.x. 
Put  y=log.  X  ;  then  u=e^y. 

\og.u=x-\-\og.y.        -_=cZa;-f--- 
u  y 

■r,   .     J       dx         T    dy         dx 
But  dy= — ,  and      ^  — 


X  y      X  log.  X 

^TTj,  du      y    .       dx  du      x\opi;.x-\-\ 

Whence,  — =dx4- = — ^ L_ 

u  x\og:.x  udx         a;locr.a; 


du (x]og.x-\-'i)u (x  log.  x-\-\  )f  log.  X 

dx  X  log.  X  X  log.  X 

(17.)     Differentiate  u=^ — - — 
Ande'+1  =  $  [       (2)  Q 

au^^JL-^^  (3) 

Differentiating  (1)  gives  e^dx^^dP.  (4) 

And  (2)  gives  e'dx^=dQ  (5) 

Whence.  (e^-f- 1  )e'dx=  QdP 

And,  {e''-^\)e'dx=PdQ 

By  subtraction,  ^e^dx=QdP—PdQ 

Whence,  du= ~-. 

21 


M  ROBINSON'S   SEQUEL. 


CIRCULAR  FUNCTIONS. 

For  the  sake  of  reference  we  will  here  note  down  the  differential 
expressions  for  trigonometrical  lines. 

Let  the  radius  of  a  circle  be  unity.  Represent  an  arc  by  x, 
then  its  differential  will  be  dx. 

d  sm.x=cos.xdx         (1)         d  cos.x= — sin.x  dx     (2) 

d  ver.  sin.a;=sin.a;G?a;  (3)         d  sec.a;=-?^ —        (4) 

cos.ar 

d  i2ing.x'=—-—      (5)         c^  cot.=— __^_       (6) 
cos.^ar       ^   ^  sin.2.r        ^    ^ 

One  great  difficulty  which  troubles  and  perplexes  the  studeiit 
in  the  calculus,  arises  from  the  fact  that  only  the  abstract  theory 
of  the  science  has  been  hitherto  brought  to  our  notice,  in  our  ele- 
mentary books. 

All  can  understand  how  these  equations,  (1),  (2),  (3),  <fec.,  are 
obtained ;  but  what  if  we  can  ?  says  the  inquiring  student. 
What  use  are  they  ?     What  do  we  learn  by  them  ? 

It  is^  useless  to  answer  these  questions  by  words  only,  we  must 
show  the  answer  by  the  following 

EXAMPLES. 

Equation  (1),  for  example,  shows  a  general  truth.  It  is  true 
applied  any  where  along  the  quadrant  of  a  circle,  x  is  any  arc 
that  we  choose  to  assume,  and  dx  must  be  an  arc  sufficiently 
small  to  be  considered  a  straight  line. 

If  a;=20°,  and  dx=l',  then  the  difference  between  the  sine  of 
20°  and  the  sine  of  20°  1',  is  d  sin.a:=cos.  20°Xl'. 

COS.  20° 9.972986 

Log.  sin.  or  arc  of  1'= .6.463726 

Sum  less  20=  log.  of  0.0002733 —4.436712 

To  the  natural  sine  of  20° 342020 

Add  the  differential .0002733 

Sum  is  the  Nat.  sine  of  20°  1' 3422933 

Thus  we  might  give  examples  without  end. 


CALCULUS.  323 

Because  the  diflferential  of  a  logarithm  is  the  differential  of  the 
quantity  divided  by  the  quantity,  therefore 

,  .         ,  d  sin.  X     COS.  x,  .      , 

d  W.  sm.  a;= -  = ax=iC,oi.x  ax. 

sin.  X        sin. a: 

This  result  corresponds  to  the  modulus  of  unity ;  for  the  modu- 
lus of  our  common  system  we  must  multiply  by  0.43429448=wz. 

For  example,  if  we  assume  ar=25°,  and  also  assume  dx-=\',  the 
differential,  or  the  difference  between  the  log.  sine  of  25°  and  the 
log.  sine  of  25°  V  is  expressed  by    m  cot.  25°  XI'- 

Log.m —1.637784 

cot.  25° 0.331327 

Log.  sine  V,  less  10 .—4.463726 

.0002709 —4.432837 

To  the  log.  sine  of  25° 9.625948 

Add  the  differential .000271 

Log.  sine  of  25°  1'= 9.626219 

We  might  assume  dx=^'  as  well  as  V,  without  error  as  far  as 
six  places  of  decimals  ;  but  it  would  not  do  to  assume  dx=  any 
large  number  of  minutes  ;  hence  the  differential  calculus  must 
be  applied  with  judgment.* 

To  show  another  example  of  the  utility  of  the  calculus,  we  will 
let  E  represent  the  obliquity  of  the  ecliptic,  L  the  longitude  of 
the  sun  at  any  time  whatever,  and  D  its  corresponding  declina- 
tion, the  radius  of  the  sphere  being  unity. 

Then  the  following  equation  is  general  : 

sin.  jD=sin. -£'sin.  X  (1) 

This  equation  represents  the  sine  of  the  sun's  declination  at  any 
point  whatever,  along  the  ecliptic.  Because  sin.  Z  is  1  at  the 
points  where  Z=90°  or  270°  ;  therefore  at  these  points  sin.  i>= 
sin.  E,  or  J)=iE,  as  it  should  be. 

Now  suppose  that  the  sun  changes  its  longitude  10',  which  we 
may  call  tlie  {dL),  or  the  differential  of  L,  what  will  be  the  cor- 

*  Those  who  are  naturally  more  nice  than  wise,  are  commonly  prejudiced 
against  this  science,  and  such  frequently  say  it  is  no  science  at  all ;  how- 
ever, their  objections  are  of  no  consequence. 


324  ROBINSON'S  SEQUEL. 

responding  change  in  its  declination,  or  wliat  will  be  the  value  of 
(dD)'! 

To  answer  this  question  we  must  take  the  differential  of  each 
member  of  the  general  equation,  then  we  shall  have 

COS.  DdD=fim.  U COS.  LdL  (2) 

^  Now  whatever  values  we  may  assign  to  L  and  dL,  equations 
(1)  and  (2)  will  always  give  D  and  dD  at  any  point. 
For  a  definite  example  we  give  the  following  : 
What  will  be  ihe  differential  in  declination  corresponding  to  the 
differential  of  10'  in  longitude  at    35°   of    longitude  ? 

In  other  words,  what  will  be  the  change  in  the  sun's  declina- 
tion'while  it  passes  from  longitude  35°  to  35°  10'  ? 
From  ( 1 )  we'  find  D  thus  : 

"^  sin.  E 9.599970 

sin.  Z  35° 9.758591 

sin.  B  13°  12'  5" 9.358561 

From  (2),  ^^^sin.  ^co.^Zr/Z 

cos.  1) 

sin.  E... 9.599970 

COS.  Z  35° 9.913365 

c?Z=  10     log 1. 

COS.  D,  complement 0.011629 

Sum  (less  20)         3.343 0.524964 

This  is  3' 20"6  nearly,  and  if  the  sun's  declination  is  13°  12' 
5",  when  its  longitude  is  35°,  the  declination  must  be  13°  15' 
25"6  at  the  longitude  35°  10'. 

This  is  nc^  strictbj  true,  because  the  ratio  of  motion  in  declina- 
tion changes  in  a  very  slight  degree  between  35°  and  35°  10'. 
But  the  ratio  between  the  motion  in  longitude  and  the  motion  in 
declination,  is  strictly  as  1  to  .3343,  at  the  beginning  of  the  arc 
between  35°  and  35°  10'. 

This  ratio,  is  constantly  changing,  but  still  equation  (2)  always 
represents  it. 

We  now  require  the  ratio  of  motion  in  declination  when  the 
sun's  longitude  is  90°,  that  is,  Z=90°  ;  then  cos".  Z=0  ;  and  sub- 
stituting this  value  in  (2)  will  cause  the  second  member  to  dis- 
appear ;  and 

cos.  D  dD=0 


CALCULUS. 


326 


Now  one  or  the  other  of  these  factors  must  be  zero  ;  but  cos.  D 
is  evidently  not  zero  ;  therefore  {dD)  must  equal  0,  showing  that 
there  is  no  motion  in  declination  exactly  at  that  point. 

On  the  contrary,  we  may  demand  the  sun's  longitude  when  the 
motion  in  declination  becomes  zero. 

In  other  words,  what  will  equation  (2)  show  when   (dD)=0  ? 

Then,  "sin.  ^cos.  L  dL=0. 

One  of  these  three  factors  must  be  zero;  but  (sin.  JE)  cannot  be 
zero,  for  it  is  a  known  constant  quantity  ;  (dL)  cannot  equal  zero  in 
case  the  least  possible  time  elapses,  for  the  sun's  apparent  motion 
never  ceases  ;  then  (cos.  L)  must  be  zero,  and  Z=90°,  or  270°. 

To  show  another  example  of  the  utility  of  the  calculus,  we 
present  the  problem  that  appears  in  another  shape,  on  page  229, 
of  our  Surveying  and  Navigation,  namely  : 

Under  ivhat  circumstances  will  an  error  in  the  altitude  of  the  sun, 
produce  the  least  possible  error  in  the  time  deduced  therefrom  ;  the 
declination  and  latitude  being  constant  quantities. 

Let  P  be  the  polar  point, 
Hh  the  horizon,  S  the  posi- 
tion of  the  sun,  and  Z  the 
zenith  of  the  observer. 

Let  A  =  the  altitude  of 
the  sun,  D  its  decHnation, 
and  X  the  latitude  of  the 
observer. 

Then  by  spherical  trig- 
onometry (see  page  214, 
Surveying  and  Navigation) 
we  have 


COS.  P= 


sin.  A — sin.  L  cos.  D 


COS.  L  sin.  D 

Tht  altitude  A  varies,  and  P  the  polar  angle,  or  time  from 
apparent  noon,  must  vary  in  consequence  of  the  variations  of  A  ; 
and^if  A  is  not  accurately  taken,  P  will  not  be  accurate. 

In  short,  a  differential  to  A,  will  produce   a  differential  in  P. 


326  ROBINSON'S  SEQUEL. 

Therefore  we  must  diflferentiate  the  equation,  taking  A  and  P  as 
variables,  and  L  and  D  constants. 

Whence,  —sin.  P  dP=.^^^-—-  ( 1 ) 

cos.L  sin.i> 

But  in  the  triangle  PZS,  we  have 

cos.  A  ^  :     sin.  P     :    :     cos.  D     :     sin.  SZP,  or  sin.  Z 

f.  .     sin.  P  COS.  D  ,a\ 

Or,  cos.  A=^ (2) 

sin.Z  ^   ^ 

Substituting  this  value  of  cos.  ^  in  (1),  and  dividing  by  sine 

P,  we  obtain 

-  dP= ^?.^;A^___x^^ 

cos.  Z  sin.  D  sin.  Z 

Or,  —dP=^- ^.^ , (3) 

cos.  L  tan.  D  sin.  Z 

Now  it  is  obvious  that  the  numerical  value  of  (dP),  or  error 
in  time,  will  be  least  when  the  denominator  of  the  fraction  in  the 
second  member  is  greatest,  and  that  will  be  greatest  when  sin.  Z 
is  greatest,  which  is  the  case  whenever  Z=90°,  which  is  when 
the  sun  is  east  or  west  of  the  observer. 

The  sign  before  {dP)  being  minus,  shows  that  when  the  altitude 
A  increases,  the  angle  P  decreases. 

If  we  make  dA^=0,  equation  (3)  Avill  give  dP=0,  showing 
that  if  there  be  no  error  in  A,  there  will  be  none  in  P. 

We  give  a  few  more  examples  in  circular  functions. 

(1.)  Let  Z  be  an  arc  or  angle  whose  radius  is  wiity  and  cosine 
(mx)  :  we  require  the  differential  of  the  arc  in  terms  ofmx. 

In  other  words,  differentiate  Z=cos.~*(77iar). 

The  rules  of  operation  are  all  comprised  in  the  following 
equations  : 

d  sin.a:=cos.  x  dx     ( 1 )         d  cos.a:= — sin.a;  dx     (2) 

"        <ftang.a;=-^     (3)         d  cot.x=—-^—      (4) 
""         cos.^a;     ^   ^  sm.^a;      ^   ^ 

in  which  x  represents  the  arc  to  radius  unity. 

Because  to  all  arcs  sin.^-[-cos.^=l.  If  cos.  =nix,  the  sine= 
Jl—m^x''. 

The  differential  of  the  first  member  of  our  equation  is  evi- 


CALCULUS.  327 

dently  dZ ;  that  of  the  second  member  is  <f(cos.-»m.r),  which  bv 
equation  (2)  gives         dZ=z    ~^^-_ 

(2.)     Differentiate     Z=sin.~^(ma;). 

If  {mx)  is  the  sine  of  an  arc  the  cosine  is  ^i — m^x^ . 

Whence,  dZ=-J^J^ 

(3.)     Differentiate  the  equation 

Z=sin.-(_X^S 

If    (  — ^- )  represents  the  sine  of  an  arc,  the  cosine  of  the 

same  arc  must  be     H      2y^ 

We  can  perform  this  operation  the  most  clearly,  by  observing 
the  following  proportion  : 

The  differential  of  any  arc,  is  to  the  differential  of  its  cosine  (taken 
negatively) y  as  radius  (or  unity)  is  to  the  sine  of  the  same  arc. 

This  proportion  is  drawn  from  the  consideration  that  if  x  rep- 
resents an  arc,  its  differential  is  dx  and  its  cosine  is  (cos. a;)  and 
the  differential  of  cos.a:  by  (2)  is  — sin.a;  dx,  and  taken  negatively, 
it  is  sin. a:  dx  ;  and  obviously 

dx     :     sin.a;  dx     :    :     \     :     sin.ar 

Applying  this  proportion  to  the  example  before  us,  we  have 

dZ     :    ^dj'^l    :   :     1     :     ^^       («) 

The  difficulty,  (all  there  is  of  difficulty),  is  in  taking  the  dif- 
ferential of  the  second  term. 


Put  JLJ^=Q;  then  (dQ)  will  be  the  differential  value 
of  the  second  term  in  the  proportion. 

Differentiating  the  first  member  as  a  fraction,  we  have 


328  ROBINSON'S  SEQUEL. 

—Aydy(\—y^  )+^ydy{  i-2y^  )_^  ^^^ 
{\—y^Y 

Reducing,   =5(iz-ll)±(il=V)=  ^ 

{^-y'Y  ydy 

That  is,       ,_:=V=^ 
(1—2/2)2      ydy 


Whence,      -c^(2=_J^_   =-_J^>/lzll_, 
By  substituting  this  value,  proportion  (a),   becomes 


rfZ 


(1— y')Vi— 2y'  VI— y' 

Or,    dZ    :     M^-y') :   :     1     :     1 

Whence,         dZ^  ^^ 


(\-.y^)  J\—Oiy^ 


(4.)     Differentiate  the  equation  Z=sin.~^(2w^l — ^m^.) 
If  2w^l — w^  is  the  sine  of  an  arc,  the  cosine  of  the  same  arc 
must  be  (1— 2«^2). 

By  the  proportion  observed  in  the  last  example, 

dZ    :     ^d{\—2u^)     :    :     1     :     ^ujv^^ 
dZ     :    '         ^udu 

Whence  dZ=. 


That  is,  dZ    :   '        ^udu  :    :     1     :     ^ujX-^w" 

9,du 


(5.)     Differentiate  the  equation  w=cos.  x  sin.  2x. 
Regarding  the  second  member  as  a  product,  and  observing  the 
dififerentials  for  sines  and  cosines,  we  have 

du= — sin.  X  sin.  2xdx-\-  2  cos.^2a;  cos.  xdx 

Whence,  _if =(cos.  2ar  cos.  x — sin.  2a:  sin.  a;)-|-cos.ar  cos.2;r. 

dx 

By  observing  equation  (9),  page   141,   Robinson's  Geometry, 

we  perceive  that  the  quantity  in  parenthesis  is  the  same  as  cos. 

(te-\-x)y  or  COS.  3a; ;  therefore, 

c?w=(cos.  3ar-}-cos  2a;  COS.  ar)c3?a;. 


CALCULUS.  389 

(6.)     Differentiate  the  equation  w=(taii.  «)". 

Put  tan.  x==y  ;  then  w=2/°,  and  duz=^ny^^dy  (a) 

But  v=tan.a;';  therefore  c/y=(^  (tan.  0?)= ,  see  (3). 

cos.^a; 

Whence  (a)  becomes  du=.  -^ '--l- 

cos.^a; 

Sin  W/j* 
(7.)     Differentiate  the  equation  u=--—-^ — ---. 

Let  sin.  nx^=P,  and  sin.  x^  Q. 

p 
Then  the  equation  becomes  w= — 

Whence,  du^^^^J^H.^^^ 

Dividing  numerator  and  denominator  by  Q'^S  and  we  have 

Because  P=sin.  nx,    dP=n  cos.  nx  dx,    and  because 
^=sin.  a;       dQ=GOS.xdx. 
Now  by  substituting  the  values  of  P,  Q,  dP  and  dQ,  (a)  becomes 

7  (n  sin.  X  cos.  nx — n  sin.  7ix  cos.  x)  dx 

("sin.  xy"*'' 
That  is,  by  equation  (8),  page  141,  Geometry^ 

,  nsin.(nx — x)dx 

"^         (ibT^'^T 

(8.)      Differentiate  the  equation  «=log.(cos.  x-\-J — 1  sin.  x. 
The  second  member  being  a  logarithm,  its  differential  is  the 
differential  of  the  quantity  divided  by  the  quantity.     That  is, 

7  — sin.  X  dx-\-J — 1  cos.  x  dx. 

J — 1  sin.a:-|-cos.a; 


Or, 


du — sin.  x-^-J — 1  COS.  a; . — r- 

^^       J — 1  sin.  ic-f-cos.ir 


Whence,  du=J — \dx. 

(9.)     Differentiate  the  equation  z<=sin. 


Jl+: 


TT 


33G  ROBINSON'S  SEQUEL. 

X 

must  be  ;  and  we  must  have 


If  the  sine  of  an  arc  is  the  cosine  of  the   same   arc 


/        1 


=) ,, 


X 


That  is,  du     :      ^dx^l+^x^     :    :     1     : 


^x 

— -     :    :     1     :     1  (;w=- 

1+x^  l-\-x^ 


Orv      ^..     :     -J^-     :    :     1     :     1  du=    ^"^ 


LUKAR   OBSERVATIONS. 

The  differential  calculus  may  be  used  with  great  facility  and 
success  in  clearing  lunar  distances  from  the  effects  of  parallax  and 
refraction. 

Let  S'm'  =  the  apparent  central  distance 
between  the  sun  and  moon,  or  the  moon  and 
a  star,  SS'  is  the  refraction  of  the  sun  or  star, 
and  it  is  sufficiently  small  to  be  taken  as  the 
differential  of  the  altitude.  Also,  m'm  is  the 
correction  for  the  moon's  apparent  altitude, 
and  we  may  call  it  the  differential  of  the  moon's 
altitude. 

The  observed  triangle  is  ZS'm'.  Let  aS^  represent  the  altitude 
of  the  sun  or  star,  m  the  altitude  of  the  moon,  and  x=S'm'y  the 
observed  distance. 

Now  by  the  fundamental  equation  of  spherical  trigonometry, 

(see  Geometry,  page  191),  we  have 

^     COS.  ir — sin. /S sin. m *        ^^k 
COS.  Z= -— (1) 

COS.  O^COS.  Wl 

We  now  take  the  differential  of  this  equation,  observing  that  Z 
is  constant,  and  that  x  varies  only  on  account  of  the  variations 
of  m  and  S, 

*  Observing  that  the  sine  of  an  altitude  is  the  same  as  tlie  cosine  of  the 
corresponding  zenith  distance. 


CALCULUS.  sfl 

First  clear  of  fractions,  tlien  differentiate  ;  then  we  shall  have 
— cos,Z(sin./S'cos.m  dS-\-cos.Ssin.m  dm)=  — sin.a;  dx — cos.S  sin. 
m  dS — cos.w  sin.S  dm. 

Changing  all  the  signs,  and  substituting  the  value  of  cos.  Z 
from  (1),  reduces  to 
/cos.arsin.  S — sin.^AS^sin.mX  ^cr  i  /cos.arsin.m — sin.^msin.^S'^ 


',\y^_./cos,x  sin.m — sin.^m  sin.^S'X  , 
/  \       •  cos.m  / 


\  cos.  S 

=sin.a;  t/ar-j-cos./S^sin.m  dS-]-CQS.7/i  sm.Sdm. 

Transposing  and  uniting  the  coefficients  of  dS  and  of  dm,  will 
give 

(cos.a;  sin. S — sin.  ^  S  sin.m — cos. ^  S  sm.m)dS 

Ccos.a:  sin.m — sin. ^m  sin. /S' — cos.^m  sin./S^  )c?m         .         , 

^ 1 —  =  sm.a;  ax 

COS.  m 

Observing  that  sin.  2 /S'4-cos.^/S'=l,  andsin.^m-[-cos.^m=l,  and 
changing  the  order  of  the  terms,  we  perceive  that 

\  cos.m  /  \  COS.  S  / 

Here  we  should  observe  that  (dm)  is  an  elevation  of  the  moon's 
apparent  altitude,  and  (dS)  is  a  depression  of  the  sun  or  star's 
apparent  altitude,  therefore  if  we  take  (dm)  positive,  (dS)  must  be 
taken  negative.     Therefore, 

^^_/c_os^^^^i^Illn^y^_/^^  (2) 

\       sin. a;  cos.m       /  \       sin. a;  cos. aS        / 

This  is  the  final  equation,  (dx)  representing  the  quantity  be- 
tween the  true  and  apparent  distance. 

Sometimes  (dx)  is  positive  and  sometimes  negative,  according 
as  the  differential  coefficient,  or  quantities  in  parenthesis  are  posi- 
tive or  negative.  When  the  differential  coefficient  of  (dS)  is 
negative,  that  term  becomes  positive,  because  (dS)  is  negative, 
and  the  product  of  two  negatives  is  positive. 

When  the  altitude  of  the  sun  or  star  is  greater  than  60°,  the 
corresponding  refraction  ((iS)  will  be  a  very  small  quantity,  which 
can  never  be  augmented  by  its  coefficient ;  therefore  in  that  case 
the  value  of  (dx)  will  be  sufficiently  represented  by 

/c_os^^n.m-sin^5y^  (3) 

\       sin.a:  cos.m        / 


% 


332  ROBINSON'S  SEQUEL. 

Equation  (2)  will  solve  any  example  that  may  be  prepared. 
We  will  solve  one  or  two  of  those  found  on  page  227,  of  our 
Surveying  and  Navigation. 

For  the  first  example  there  found,  in  which  S=Q6°  3',  m=39° 
18',  x==46°  45',  and  the  mpon's  horizontal  parallax  53'  51",  ex- 
pression (3)  will  be  sufficient. 

We  must  first  find  (dm),  which  is  the  parallax  in  altitude 
diminished  by  the  refraction. 

53'  51  "=3231"     log 3.509337 

cos.  m  39°  18' 9.880651 

log.  2499" 3..397988 

39°  18'    Refraction,...      69" 

2430"=(Zm 
For  the  coefficient,  we  operate  thus  :  (radius  unity.) 

sin.  m.. — 1.801665     cos.  7W.  .—1.888651 
cos.  a;.. — 1.835807     sin.  a:... — 1.862353 
log.  4340    ..—1.637472  —1.751004 

Nat.  sin.  aS'         99762  [sub.  the  upper.] 

-56362*     log — -1 .750975 

^—1.999971 

c?m=2430     log -.3.385606 

dx=40'  29"=.2429" 3.385577 

X  orthe  app.  dis.  46°  45' 
True  distance  46°    4'  31" 

The  answer  in  the  book  is  46°  4'  25".  Our  omission  of  the 
second  term  in  equation  (2)  might  have  produced  an  error  of  4", 
not  more,  still  making  a  difference  of  2",  but  this  is  of  no  conse- 
quence in  itself.  Different  operators  may  work  the  same  exam- 
ple by  the  same  or  different  methods,  and  they  will  rarely  produce 
results  within  5"  of  each  other ;  and  as  no  observations  can  be 
relied  on  within  that  limit,  such  results  in  a  practical  point  of 
view  are  said  to  agree. 

We  now  take  the  6th  example  from  page  227,  Surveying  and 
Navigation. 

*  Because  this  quantity  is  minus,  (dx)  must  be  minus,  and  therefore  we 
subtract  it  from  the  apparent  distance  to  find  the  true  distance. 


CALCULUS.  3to 

Given  sun's  app.  alt.  8°  26',  Q)'s  app.  alt.  19°  24',  horizontal 
parallax  67'  14".  Apparent  central  distance  120°  18'  46",  to  find 
the  true  distance.  Ans.   120°  1'  46".. 

Here  S=^°  26',  m=19°  24',  and  ar=120°  18'  46". 

Horizontal  parallax  57'  14"=3434" 3.535800 

cos.m     19°  24' 9.974614 

Parallax  in  altitude    3239" 3.51041 4 

Refraction ^^    161" 

dm=: .  .T:T3078"  dS=:6'  10"=370". 

(Because  x  is  greater  than  90°  its  cosine  will  be  minus,  which 
will  render  the  differential  coefficient  of  dm  minus.) 

sin.m... — 1.521349         cos.m... — 1.974614 
COS.  a;...— 1 .703045         sin.  a: —1.936152 

—.16763  —1.224394  —1..9 10766  den. 

Bin.S        .14666 

—.31429         log — 1.497340  num 

--1. 586574 

c?m=3078"      log 3.488269 

First  term  of  (dx) —1188" 3.074843 

sin.  S. .  .—1.166307          cos.  S .  .  .  —  1.995278 
cos.  ar. .  .—1.703045         sin. .r —1.936152 


—.07402  —2.869352  —1.931430 

sin.wi       .33216 

—.40618 -1.608736 

—1.677306 

— c?iS^=370" 2.568202 

Second  term  of  (dx) +  176" 2.245508 

—1188" 


dx= —1012"=  16'  52" 

Apparent  central  distance, 120°  18'  46" 

True  distance, 1 20°    1'  54" 

The  result  differs  8"  from  the  given  answer  as  determined  by 
other  methods,  which  arises  from  taking  3078"  as  a  differential ; 
it  is  a  large  arc,  rather  too  large  to  be  taken  for  a  differential  arc. 


334  ROBINSON'S  SEQUEL. 

MAXIMA  AND  MINIMA. 

The  differential  of  any  quantity  is  a  general  expression  for  a 
small  increase  of  the  quantity  ;  but  if  the  quantity  is  already  a 
maximum,  an  increase  is  impossible,  and  the  expression  for  an  in- 
crease must  be  zero. 

A  decreasing  differential  is  a  general  expression  for  a  small 
relative  decrease  of  any  quantity  ;  but  when  the  expression  is 
already  a  minimum,  it  can  have  no  further  decrease,  and  the 
expression  for  such  decrease  must  therefore  be  zero.  Hence,  in 
cases  of  a  maximum  or  minimum,  we  must  put  the  differeniial 
of  the  quantity  equal  to  zero,  thus  forming  a  new  equation,  which 
equation  generally  gives  the  results  sought. 

For  example,  the  following  equation  always  unites  the  declina- 
nation  of  the  sun  with  its  longitude : 

sin.  i)=sin.  E  sin.  L  ( 1 ) 

Here  E  is  the  obliquity  of  the  ecliptic,  and  is  a  constant  quan- 
tity. L  is  any  longitude,  and  D  the  corresponding  declination. 
Taking  the  differential  of  this  equation,  we  find 

COS.  D  dl)=sin.  E COS.  LdL  (2) 

If  we  now  assume  the  condition  that  D  is  a  maximum,  it  is  the 
same  as  to  assume  that  (dD)=0,  which  makes 
sin.  E  COS.  L  dL=0 

Here  is  the  product  of  three  factors,  one  of  which  must  equal 
0.  Sine  E  is  of  known  value,  not  equal  to  zero  ;  therefore  cos. 
LdL=0.  If  cos.Z=0,  Z=90°.  If  dL=0,  L=0.  Substitu- 
ting these  values  of  L  in  equation  (1),  we  have  sin.  i>=sin.  ^, 
or  JJ=E,  when  D  is  a  maximum,  and  sin.  i>=0,  or  D=0,  when 
i>  is  a  minimum  ;  which  are  obvious  results. 

Again,  the  general  value  of  the  differential  of  the  sine  of  any 
arc  whose  length  is  x  and  radius  unity,  is  cos.  x  dx. 

But  if  the  sine  is  a  maximum,  it  can  have  no  differential,  ex- 
cept in  form.  That  is,  cos.  x=0,  or  dx=0  ;  whence  a:=90°,  or 
x=0.  Showing  that  when  the  arc  is  90°,  the  sine  is  a  maximum, 
"when  0,  the  sine  is  0,  a  minimum. 

For  another  iWMsirsiiion,  I  propose  to  divide  the  number  a  into 
(wo  suck  parts  that  ike  product  of  tke  parts  skall  be  the  greatest  pos- 
sible.       •  K 


CALCULUS.  335 

At  first  I  will  simply  get  an  expression  for  any  indefinite  rect- 
angle.— That  is,  if  x=  one  part,  (a — x)  will  equal  the  other  part, 
and  (ax — x^)  will  be  an  expression  for  a  rectangle  which  will  be 
larger  or  smaller  according  lo  the  relative  values  of  x  and  a. 

Taking  the  differential  of  the  expression,  we  have  adx — 2xdx. 

Now  if  I  assume  (cix — x"^)  to  be  a  maximum,  it  cannot  in- 
crease, therefore  its  differential  must  be  zero,  or 

adx — 2xdx=^0.     Or,  a — 2a;=0,  or,  x=^a, 
which  is  the  value  of  x  when  the  product  is  a  minimum,   and 
may  be  verified  by  trial. 

EXAMPLES. 

(1.)  Required  the  greatest  rectangle  that  can  he  inscribed  in  the 
quadrant  of  a  given  circle. 

It  is  evident  that  one  extremity  of  the  diagonal  of  the  rectangle 
must  be  at  the  center  of  the  circle,  and  the  other  extremity  at 
some  point  on  the  arc  of  the  quadrant. 

Let  a  =  the  diagonal  or  radius  of  the  circle,  and  x  =  the  arc 
from  one  extremity  of  the  quadrant  to  the  point  in  which  the  rect- 
angle meets  the  curve.  Then  a  sin  .r=  one  side  of  the  rectangle : 
acos.  ar=  the  adjacent  side,  and  the  area  of  the  rectangle  is 
a^sin.  iccos.  x. 

The  problem  requires  that  this  expression  should  be  a  maxi- 
mum, which  is  the  same  thing  as  requiring  that  its  differential 
expression  should  be  zero. 

Hence,         a'^cos.  x  dx  cos.  x — a^sin.  x  dx  sin.  a;=0. 

Dividing  by  a^dx,  and  cos.^a; — sin.^;i:=0. 

Or,  COS.  a;=:sin.  a;. 

But  the  arc  which  has  its  cosine  equal  to  its  sine  is  45*^,  which 
shows  that  the  diagonal  of  the  rectangle  bisects  the  quadrant, 
and  the  rectangle  is  in  fact  a  square. . 

We  observe*  that  a^  disappears  in  the  result,  and  this  shows 
that  the  problem  is  independent  of  the  radius,  and  equally  ap- 
plies to  all  circles. 

In  short,  constant  factors  in  a  maximum  may  he  cast  out  hy  divi- 
sion hefore  we  take  the  diferential.  ' 

(2.)  Required  the  greatest  possible  rectangle  that  can  he  inscribed 
in  a  {fiven  parabola. 


m 


336  ROBINSON'S   SEQUEL. 

Put  VD=a,  VB=x,  PB=.y.  Then 
BD=a — X,  2t/(a — a:) = maximum,  and  y^ 
=:  2px,  by  the  equation  of  the  curve. 

Taking  the  differentials,  we  have 

dy{a^x)—ydx=0.     Or,  dy^^^ 

a — X 

ydy=pdx.     Or,  dy^=^^J^ 

y 

Whence,  ^    =±.     Or,  y-=iap — px. 

a — X      y 

That  is,  2px^=ap — px.     9.x=a — x.     x^^a. 

This  result  shows  that  from  the  vertex,  J-  of  the  given  distance 
is  the  point  through  which  to  draw  one  side  of  the  maximum 
rectangle. 

^   (3.)     Problem  3  on  page  253  of  this  work,  is  a  beautiful  exam- 
ple to  show  the  power  and  utility  of  the  calcidus. 

In  fact  it  was  the  result  of  the  calculus  that  pointed  out  the  geo- 
metrical construction. 

Let  AP=a,  PB=^b,  PD=x,  and  call  the  angle  APD=P. 

Then  I)JI=xsm.P,  PB'=xcos.P,  All^a—x  cos.  P,  HB= 
X  COS. P±ib,  according  as  IT  falls  between  A  and  B,  or  between 
B  and  P.     Corresponding  to  our   figure,  HB=x  cos.P — b. 

In   the  triangle  AHD,  we  have 

1     :     i2.ii.ADH 

1        4.         AnzT     « — ^  COS.P 
1    :  isin. AI>Ii= 


DH 

:     HA 

Or, 

X  sin 

.P  :  a- 

—X  cos.P 

Also, 

icsin 

.P  :  X  cos.P— b 

1    :  tsin.  ffDB= 


X  sin.P 
X  cos.P — b 


X  sin.  P 

Adding  these  two  tangents  according  to  the  mathematical  law 

of  the  sum  of  tangents,  expressed  in  equation  (28),  page  143  of 

Robinson's  Geometry,  will  give 

,  a — b 

X  sin.P 

tan.  ADB= 5-^ 5 — 7\ 

^ a — X  cos.P /x  cos.P — b  \ 

X  sin.P     \    X  sin.P    ^ 
(a — b)  X  sin.P 


ar^sin.^P — (a — x  cos.P)  (;c cos.P — b 


CALCULUS.  337 


t^n.ADB=—^-^^J^^''J^^ 


x"-  sin.2P+a6— (a4-6)  x  cos.P-f  a;^  cos.^P 

(a — i)a;sin.P 

x^-\-ab — (a-f-^)  a:  cos.P 
By  the  requirement  of  the  problem,  the  angle  ADB  must  be  a 
maximum  ;  but  the  angle  will  be  a  maximum  when  its  tangent  is 
a   maximum.      Hence  we  must  put  the  differential  of  the  ex- 


-^ 1 equal  to  zero. 


pression 

x'^-\-ab — (ci-\-b)  X  cos.P 

By  omitting  the  constant  factor  (a — b)  sin.  P,  and  dividing  nu- 
merator and  denominator  by  x,  we  shall  have  only 

1       

r+— — (a+i)  cos.  P     to  differentiate. 

— dx-\- — dx 
^x^ 
Whence.  -; ; " r-^=0. 


(x-{J'l-^{a-Yb)  cos.  pV 


Or,  a;2  =«6. 


This  equation  directed  us  to  make  FD  =  J 500 -200  as  was 
done  on  page  263. 

(4.)  The  difference  of  arc  between  the  sun's  right  ascension  and  its 
longitude  gives  rise  to  one  part  of  the  equation  of  time.  What  is  the 
sun's  right  ascension  when  this  part  of  the  equation  is  a  maximum,  and 
what  is  the  maximum  value  ? 

Let  Z=  the  sun's  longitude,  x=.  the  corresponding  right  ascen- 
sion, ^=  the  obliquity  of  the  echptic  ;  then  by  spherical  trigo- 
nometry we  have 

I'cos. -£'=:cot.  Z  tan.  a:= — 1-  fl) 

tan.Z  ^   ^ 

(See  equation  (16)  Robinson's  Geometry,  page  186,  also  (5),  page  139.) 

But,  tan.(Z-^)  =  A^"^i^:i  (2) 

^  ^     1+tan.Ztan.a;  ^   ^ 

(See  page  143,  equation  (29),  Geometry.) 

The  problem  requires  that  the  differential  of  {L — x)^  or  of 

tan.  (Z — x)  should  be  put  equal  to  zero. 

22 


338  ROBINSON'S  SEQUEL. 

Substituting  the  value  of  tan.  L  taken  from  (1)  in  equation  (2), 

we  have 

tan. «      . 
— tan.  a; 

cos.^  (1— cos.^)tan.a; 

tan.  li/— a;)= -: — ^ =^^ 

j  I  tan.^a;  cos.  ^+tan.2a? 

COS.  ^ 

1^,  tan.  X  .  1 

Whence, =maximum,  or 

COS.  jE'-j-tan.^a;  cos. -£'cot.a;-J-tan.a; 

=  maximum. 

But  this  fraction  is  obviously  a  maximum  when  its  denomii).ator 
is  a  minimum  ;  therefore, 

cos.  ^  cot.  a:-|-tan.  a;=minimum. 

Taking  the  differential,  we  find 

— COS.  JEJdx,     dx        ^  sin.^ar  j-, 
-j- =0                   — —  =  cos.  -A 

sin.^a:  cos.^a;  cos.^a; 

tan.  a:=^cos.  ^  (3) 

»     This  value  of  tan.  x  put  in  ( 1 )  gives 

tan.Z  =  >/^^  (4) 

cos.  ^ 

Equations  (3)  and  (4)  correspond  to  radius  unity,  but  we  can 
use  the  logarithmic  table  if  we  add  10  to  the  index  of  cos.  ^  be- 
fore we  take  the  root,  thus  : 

jE'=23°  27'  30"     cos.  +10 19.962535  (  2 

tan.  re  43°  45'  50" 9.981267 

9.962535 

tan.  Z  46°  14"  10" 10.018732 

Diff.  of  arc  2°  28'  20"=9m.  54.6s. 

(5.)      What  must  he  the  inclination  of  the   roof  of  a  huUding 
that  the  water  will  run  off  in  tlve  least  possible  time  ?       Ans.  45°. 
Let  a=  the  base  of  a  roof,  x=  its  perpendicular  altitude  :  then 

Ja^+x^  will  equal  its  length,  and  J^  will  equal  the  time  re- 
quired for  water  to  fall  through  the  height. 

But  the  time  down  an  inclined  plane  is  to  the  time  through  its 
perpendicular,  as  the  length  of  the  plane  is  to  its  height. 

Let  <=  the  time  down  the  plane  ;  then 


CALCULUS, 


339 


4 


Or, 


^=. 


J9^ 


But  the  problem  requires  that  ^  should  be  a  minimum  ;  its 
square  must  therefore  be  a  minimum  ;  hence, 

— ! =  a  mmimum. 

X 

It  will  be  observed  that  ^  is  the  force  of  gravity,  and  being  a 
constant  factor  in  the  last  expression,  it  was  omitted. 
Taking  the  differential,  we  have 

'ix^  dx—dx{a^  J^x""  )_^ 

x^=:a^^     -or  a;=a. 
The  perpendicular  being  equal  to  the  base,  shovrs  that  the  in- 
clination required  must  be  45°. 

(6.)  Within  a  triangle  is  a  given  point  P,  the  distance  to  the  nearest 
mngle  A  is  given,  and  the  line  AP  divides  the  angle  A  into  two  an- 
gles m  and  n,  of  which  m  is  greater  than  n. 

It  is  required  to  find  the  line  EF  drawn  through  the  point  P,  so 
that  the  triangle  AEF  shall  he  the  least  p>ossiMe, 

Let  AP==a,  AF=x,  AE=y.    The 
angle  PAF=^m,  FAE=^n. 

The  area  of  the  A  ABF=ax  sin.  m. 

The  area  of  the  A  APF=ag  sin.  n. 

By  the  conditions, 

ax  sin.  m-|-ay  sija.  ?*=  minimum* 
^  Also,  by  the  conditions, 

arysin.  (m-\-n)=  minimum. 

From  tlie  first  minimum, 

sin.  mdx-^sm.  ndg:=^0 

From  the  second,        xdg-^-gdx^O 

From(l), 


(1) 

(2) 


From  {2}, 


sin.m 
dx=:= — -dy 

y 


340  ROBINSON'S   SEQUEL. 

Whence,  y  sin.  n  =  x  sin.  m. 

Or,  ay  sin.  n=.ax  sin.  m. 

This  last  equation  shows  that  the  line  EPF  mustjbe  so  drawn 
that  the  triangle  APF  will  be  equal  to  the  triangle  APE. 

We  presume  that  the  foregoing  examples  are  sufficient  to  illus- 
trate the  power  and  utility  of  the  calculus  in  respect  to  maxima  and 
minima. 


INTEGRAL    CALCULUS. 

The  integral  calculus  is  the  converse  of  the  differential.  In 
the  differential  cakulus  we  give  the  integral  and  require  the  dif- 
ferential. In  the  integral  calculus  we  give  the  differential  quan- 
tity, and  require  the  integral  from  which  the  dijBferential  was 
derived.  Hence  all  our  rules  of  operation  msust  have  reference  to 
the  differential  rules  inversed. 

We  cannot  investigate  the  rules  of  operation  in  this  work,  but 
suppose  the  reader  already  acquainted  with  them. 

Many  persons  can  operate  ta  some  extent  in  the  integral  cal- 
culus without  any  distinct  idea  of  what  the  integral  calculus  is, 
and  this  vagueness  can  never  be  fully  driven  away,  except  by 
close  attention  to  the  application  and  utility  of  the  science. 

For  example. — If  we  have  the  differential  of  a  circular  arc,  by 
integration,  we  shall  have  the  arc  itself. 

If  we  have  the  differential  o-f  a  circular  segment,  by  integration 
we  shall  have  the  segment  itself. 

If  we  have  the  differential  quantity  of  a  cone,  by  integrating 
that  quantity  we  shall  have  the  solidity  of  the  cone  itself. 

If  we  have  the  differential  of  a  logarithm,  by  integration  we 
shall  have  the  logarithm  itself. 

Thus  we  might  go  through  the  chapter. 

Because  the  integral  is  the  opposite  of  the  differential,  the  inex- 
perienced might  conclude  that  one  operation  would  be  obvious 
from  the  other. 

In  some  instances  the  converse  is  obvious,  but  not  generally  so. 
We  must  not  conclude  that  it  is  as  easy  to  move  in  one  direc- 
tion as  in  the  opposite. — It  is  not  as  easy  to  ascend  as  to  descend 
a  plane — not  so  easy  for  a  vessel  to  move  against  the  stream  as 


CALCULUS.  341 

with  it.  It  is  not  so  easy  to  find  the  cube  root  of  a  number  as  it 
is  to  cube  the  root  when  found. 

The  sign  for  the  differential  is  (d).  The  sign  for  integration 
is  (  f).  Hence,  J'du=u.  The  two  signs  destroy  one  another, 
and  give  the  quantity  u. 

If  we  take  the  differential  of  a;*  we  shall  have  4x^dx.  There- 
fore if  we  must  integrate  4x^dx,  we  must  frame  a  rule  of  opera- 
tion that  will  give  x^^y  which  is  the  following  : 

Add  unity  to  the  index,  divide  hj  the  index  so  iTicreased,  and  take 
away  dx. 

The  differential  of     -  is ^  conversely  then  the 

the  integral  of is  .     But  to  integrate  this  by 

the  above  rule  appears  at  first  sight  impossible,  nevertheless  it 

can  be  accomplished  by  the  following  artifice  : 

Put         1 — x^=y.     Then  xdx=^ — \dy. 

A    J                   xdx  dy  1   -1-7 

And, _=_-_JL=_iy  Uy 

To  the  second  member  of  this  equation  the  rule  will  apply. 
Hence,        J — ^ _=y~2__ — ______  .     ^^2^^ 

The  differential  of  xy  is  {xdy-\-ydx)  ;  therefore, 

J {xdy-\-ydx)=^xy .      But  the  inquiry  is,  how  we  shall  effect 
the  integration  under  the  rule. 

Here  y  is  equal  to,  or  greater,  or  less  than  x.  Therefore  we 
may  assume  that  y=ax,  a  being  a  constant  quantity.  Whence, 
dy=:-adx,  xdy=axdx. 

And,  f  (xdy-\-ydx)=  C  (axdx-\-axdx)=:  f^axdx==^ax^. 

But  ax^=ax'X,  and  because  y=ax,     ax^  =xy.     Ans. 

Then  we  perceive  that  the  actual  integration  was  performed  by 
the  rule  ;  but  the  reader  must  not  infer  that  the  rule  will  apply  to 
all  cases — -far  from  it.* 

*  The  subject  of  integration  requires  the  keenest  algebraical  talent,  and  fe"W 
persons  are  skilful  algebraists  in  the  highest  sense  of  that  term,  who  have 
aot  been  severely  disciplined  iu  integration. 


»  f 


342  ROBINSON'S  SEQUEL. 

We  give  a  few  examples  under  this  rule. 

(1.)      Given  the  diferenilal     {^x^ydy -\-^^^xdx),  to  find  its 
i$Uegr(d.  Ans.  x^y^ . 

Here  we  may  assume  y=£ax ;  then  dyz^etdx,  ydy=.a^xdx. 
Whence,    J  (  ^"^ ydy-\-^y^ xdx)  =J'  (  2a-x^dx-{-2a''x^dx)  = 

C  Aa^x^dx. 
To  the  second  member  the  rule  applies  ;  that  is, 

C  Aa^x^dx=a^x'^=za^x^  ^x^z=iy^x^ .  Ans. 

(2.)     Inte^e^te    (J!+y'i^+(^!+^--M 

Ans.  JV+^  Ja^'^^. 
Assume   j¥^+^=P,  and  Ja^+P==Q. 
Then  ydy=FdF,  and  xdx=QdQ. 

Substituting  these  values,  and  the  given  expression  becomes 
F^QdQ-j-PQ^dP 

That  is,  PdQ-\-QdP. 

'  But,  J{PdQ-{-QdP)=PQ. 


That  is,  J^^+y^  X  Ja'^+x'' .  Ans. 

(3.)     Inte^mte  _— 3^y+3c/^  ^^^    (a— y+0)^ 

Put  (a_^-|^s)*=zP  ;  then  a— .y+2=/'^ 

And  —dy+dz=4P^dP. 

Whence.  __3^+3^  =3P^  JP. 

4(a-y+z)^ 
And,  r_Z±.^i'±?^=P3=(«-y+2)'.   ^«*. 

Observatton.     The  differential  of  -  is  VJ^^J  .    therefore. 


y  y2 


.     X 


the  integral  of  this  last  expression  is  —     But  how  shall  we  inte- 

y 

grate  this,  provided  we  did  not  know  the  integral  ? 

The  numerator  would  be  the  differential  of  the  product  xy,  if 
the  sign  between  the  terms  in  the  numerator  were  plus. 


CALCULUS.  843 

Let  us  put  y^=ax.     Then  ydx=:^axdx,  and  xdi/=axdXf  and  the 

yclx — xdv  ,  ax  dx — ax  dx  0  j  /. 

expression  f-— ^  becomes or ,  and  our  ef- 

^  y^  a^x^  a^x^ 

fort  fails.      Now  let  us  examine  the  cause  of  the  failure.     The 

product  xy  can  represent  any  magnitude  whatever,  and  if  we 

put  y=ax,  then  xy  becomes  ax^  ;  and  because  x  is  variable,  ax^ 

is  still  capable  of  representing  any  magnitude  whatever.     But  iu 

the  fractional  expression  -,  if  y=-ax,  and  a  be  regarded  as  con- 

y 

V  X  \  X 

stant,  _  becomes  — ,  or  _,  and  in  that  case  _  can  only  represent 
y  ax         a  y 

*  1  X 

the  ever  constant  fraction  -  ;  but  -  must  be  capable  of  repre- 
a  y 

senting  any  fraction  whatever  ;  therefore  we  cannot  put  y  =  ax, 

unless  we  regard  a  as  variable. 

Therefore  to  integrate  the  expression  tL_^ ?L ,  put  y  =  tx  ; 

both  t  and  x  being  variable. 

Then    ydx=ixdx,  dy:=tdx-\-xdt. 

xdy=txdx-\-x  ^  dt. 
Whence,  ydx — xdy=. — x^dt 

J  ,,  .      r  x^dt  dt 

and  the  expression  becomes      — or— — 

t  X  z 

That  is,  jyl^HpL^J^t-^dt^t-^  hythe  rule. 
Whence,  the  required  integral  is  -  ;    but  y  ^=ix  ;   therefore, 

t    y 

This  branch  of  the  subject  may  be  treated  as  follows,  provided 
the  operator  is  cautious,  and  does  not  assume  too  much  : 

When  we  differentiate  a  product  as  xy,  we  assume  x  as  con- 
stant and  y  variable  ;  and  then  y  constant  and  x  variable,  and 
thus  we  get  two  partial  diflferentials. 

Now  either  one  of  these  integrated  on  the  supposition  thai  the 
letter  which  is  affixed  to  (^d)  is  the  variable  one,  and  all  others  con- 


344  ROBINSON'S  SEQUEL. 

stant,  will  give  the  true  integral.      Thus  the  diflferential  of  xy  is 

xdy-\-ydx. 
Now  integrate  xdy  on  the  supposition  that  x  is  constant  and  y 
variable,  and  we  have  xy  for  the  integral.    ^Iso,    Cydx=xy.    It 
would  therefore  appear  that  ^xy  is  the  whole  integral,  provided 
we  did  not  know,  a  priori,  that  xy  is  the  integral. 

ffence,  when  we  integrate  two  differential  expressions,  on  the  sup- 
position that  the  letter  not  affected  with  the  differential  sign  (d),  is 
constant,  and  find  two  equal  integrals,  we  must  take  but  one  of  them. 

The  same  principle  holds  good  in  relation  to  the  three  or  more 
letters.     The  diflferential  of  xyz  is 

xydz-\-xzdy-\-yzdx. 

Now  if  we  integrate  this  expression  on  the  supposition  that  xy 
is  constant  in  the  first  term,  xz  constant  in  the  second,  and  yz 
constant  in  the  third,  we  shall  have 

xyz-\-xyz-\-xyz. 
Here  are  three  equal  integrals,  but  we  must  take  but  one  of  these 
for  the  whole  integral,  because  the  differential  was  eflfected  by 
three  distinct  suppositions. 

The  diflferential  of  -  is  yJf^Z^l^J^—xy-^dy, 

y         y""  y 

Integrating  each  of  these  expressions  on  the  supposition  that  y 
is  constant  in  the  first,  and  x  constant  in  the  second,  we  have 

x.x 

y    y 

but  we  must  only  take  one  of  these  for  the  integral,  for  the  same 
reason  as  before. 

EXAMPLES. 

(1.)     Integrate        (Sxy^y'^  )dx-\-(3x^ —'2xy)dy 

Ans.  Sx^y — y^x. 
We  integrate  the  first  part  on  the  supposition  that  y  is  constant, 
and  the  second  on  the  supposition  that  x  is  constant,  and  we  obtain 

3x^ij—y^x-{-3x^y—y^x, 
and  because  we  make  two  distinct  suppositions,  we  divide  by  2. 
Then  test  the  result  by  taking  the  diflferential. 


CALCULUS.  346 

(2.)     Integrate        {'iy''x-\'^^)dx-{-{^x^y-\-^xy'^'\'^y^)dy. 
Integrating  each  term,  we  obtain 

y^x^-{-Sy^x-\-x^y^^^xy^-\-^y\ 
Here  we  find  two  terms  equal  to  aj^y^,  and  two  terms  equal  to 
^xy^f  and  one  term  Sy*  ;  hence  I  will   take 

for  the  integral  sought — which  I  find  to  be  true  by  taking  the 
differential. 

To  integrate  the  varied  expressions  in  the  form  ' 
x'^{a-\-bx'^ydx, 
we  must  resort  to  the  established  formulas,  explained  in  elaborate 
works,  which  of  course  we  cannot  touch  upon  in  a  work  like  this. 

Because  c?  log.  ar= Therefore,       f =.\og.x-\-c     {a) 

X  ^      X 

**        d  sin.  x=cos.xdx.        "  f  cos.  xdx=sm.  x-\-c  (b) 

**        d  cos.  a;= — sin.  xdx.    "  C — sin.icc?ar=cos.a;-j-c  (c) 

'*       .<^tan.x=_^-.       »  **  r_^_=tan.a;+c     {d) 

cos.^a;  *^  cos.^a; 

Each  of  these  formula  is  a  fundamental  rule  for  integration. 
It  is  not  necessary  for  us  to  explain  the  constard  c. 

APPLICATION"  OF  THE  INTEGRAL  CALCULUS. 

We  shall  explain  the  application  and  utility  of  this  science  by 
examples. 

If  x  represents  an  arc  of  a  circle  whose  radius  is  unity,  and  y 
the  sine  of  the  same  arc  ;  then  ^1 — y'^  will  represent  the  cosine, 
and  equation  {b)  above  will  become 

The  integral  of  the  first  member  will  give  the  arc^  but  it  will 
be  numerically  indefinite,  unless  we  can  integrate  the  second 
member,  and  know  the  value  of  y  corresponding  to  some  definite 
value  of  X. 


346  ROBINSON'S  SEQUEL. 

We  cannot  integrate  the  second  member  in  finite  terms,  there- 
fore we  must  develop  it  in  a  series,  and  integrate  term  by  term, 
and  if  the  series  is  suflficiently  converging,  the  value  of  x  can  be 
known  to  any  required  degree  of  approximation. 

'  — L 

:=(1 — y^)   ^dy.      The  binomial,  expanded  by  the  bi- 


/I- 

nomial  theorem,  produces 

Multiplying  each  term  by  dy,  and  integrating,  we  obtain 

,  l-y3  ,  1   3  ys  ,  1    3  5  y^  ,  1   3  5  7  v%    «       , 
'2-3      245'2467'24689'  ' 

This  equation  is  true  for  all  values  of  x.  It  is  true  then  when 
a;=0  ;  but  if  we  make  a:=0,  y  must  equal  0  at  the  same  time. 
Therefore,  if  we  make  the  supposition  that  x=0,  the  last  equa- 
tion will  become  0=0-|-c,  or  c=0.  By  some  such  special  con- 
sideration, the  value  of  c  can  be  determined  in  almost  every 
problem,  although  it  is  indeterminate  in  the  abstract. 

Now  the  value  of  y  is  known  to  be  \  when  x,  the  arc,  equals 
30°;  therefore. 

The  arc  of  30==l+iA+llll+M-l-l+^Li:^lI^_ 
2  '  3.2*  '  4.5.2«  '  4.6.7.28  '  4.6.8.9.2»'> 

+  &C. 

Multiplying  by  6  and  taking  ten  terms  of  the  series,  we  shall 
have  the  value  of  a  semicircle  to  radius  unity.  That  is,  we  shall 
have 

7t=3.1415926, 

which  is  true  to  the  last  figure. 

Thus  we  perceive  that  one  operation  in  the  integral  calculus 
brings  a  result  requiring  many  operations  in  common  geometry. 

SURFACES    AND    SOLIDS. 

In  general  terms,  aydx  will  represent  the  differential  of  any 
plane  surface,  and  if  so,    Caydx-\-c,  will  represent  any  surface  ; 


I 


CALCULUS.  347 

but  we  can  find  the  integral  only  when  we  know  some  relation 
between  x  and  y. 

Also,  ay^dx  will  represent  the  differential  of  any  solid  ;  there- 
fore Cay^dx-\-c  will  represent  the  solid  itself ;  but  we  can  find 
this  integral  only  when  we  have  some  relation  between  x  and  y. 

EXAMPLES, 

Suppose  x  to  represent  the  perpendicular  of  any  triangle,  and 
y  its  base  ;  then  if  x  increases  downwards  by  dx,  ydx  will  be  the 
differential  of  the  triangle,  the  angles  remaining  constant. 

Therefore,  Cydx  will  be  the  area  of  the  triangle  itself.  This 
integral  will  require  no  correction,  for  when  x=^0,  y=0. 

The  area  being  a  triangle,  we  have  a  relation  between  x  and  y, 
for  no  triangle  can  exist  without  this  numerical  relation. 

Suppose  we  measure  one  unit  down  the  base,  and  through  that 
point  draw  a  line  parallel  to  the  base,  and  find  the  length  of  this 
to  be  a  units.  Then  whatever  be  the  magnitudes  of  x  and  y,  this 
relation  will  be  constant,  and  a  will  be  greater  or  less  according 
to  the  angles  of  the  triangle. 

That  is,         X     :     y     :    :     1     :     a.         Or,  y=ax. 

Consequently,        Cydx^zfcixdx^ =— .ar=:-iL 

That  is,  the  area  of  any  triangle  is  half  the  product  of  its  base  and 
altitude. 

Let  VCI  be  any  area.       VC=  x,    CI=y, 

CD  =  dx,  then  the  space  CDRI=i  ydx,  the 

differential  of  the  area. 

If  VCI  represents  a  portion  of  a  parabola, 
1.  I 
then   y^=z<ipx.     Or,  ?/=(2^)2a;2. 

Whence,  Jydx=J{'^pYx^dx=%(9.pyx^. 

But  Ty=(<ipyx^;  therefore  VCIi^^VOIg. 
If  VCI  is  a  portion  of  a  circular  area  of  which  r  is  the  radius, 
.     Then         (^r--^Y -\-y^ ^^r^ ,  and   y^^^Sra;— a:^. 
Or.  y=sj^rx — x^. 


348  ROBINSON'S  SEQUEL. 


Hence,    fydy=f  J^rx — x^dx=.  the  area  of  this  semi-segment. 

We  cannot  integrate  this  in  finite  terms,  Ave  can  only  approx- 
imate to  the  integral  by  expanding  the  binomial,  multiplying  each 
term  by  dx,  and  then  integrating  each  term  separately. 

After  integration,  if  we  take  a;=2r,  the  result  will  be  the  area  of 
a  semicircle  whose  radius  is  r. 

SOLIDS. 

(1.)     To  find  the  solidity  of  a  cone. 

If  X  represent  the  perpendicular  altitude  of  an  upright  cone  and 
y  the  radius  of  its  base,  then  Tty^  will  equal  the  area  of  the  base, 
and  if  x  be  increased  by  dx,  rty^dx  will  be  the  differential  of  the 
cone.     Consequently,  J'Tty^dx  will  be  the  solidity  of  the  cone. 

As  any  cone  whatever  must  have  some  constant  ratio  between 
its  perpendicular  and  base  ;  therefore, 

x     :    y     :    :     \     :     a.     Or,  y=ax. 

Whence,    CTty^dx^::  C7ia^x^dx=^^Tta^x^=:^x'7ta^x^=\x-7<.y^ . 

That  is,  the  solidity  of  a  cone  is  equal  to  the  area  of  the  base  mul- 
tiplied by  one-third  of  the  altitude. 

N.  B.  The  integral  required  no  correction  because  x  and  y 
vanish  together. 

(2.)     To  find  the  solidity  of  a  paraboloid. 

Let  VCI  revolve  on  the  axis  VC  (see  last  figure,)  it  will  de- 
scribe a  paraboloid  of  which  jty^dx  is  the  difi'erential.     y^=^2px. 

Therefore,       Cny^dx^  C^7tpxdx=pTCX^ . 

To  form  a  correct  idea  of  this  solid,  we  must  observe  that  jty^ 
z=.^piix.  Consequently  2/)7ta;  is  the  area  of  the  circle  described 
by  the  revolution  of  y,  and  therefore  ^pnx'  is  the  solidity  of  the 
cyhnder  which  would  just  circumscribe  the  paraboloid,  and  hence 
we  perceive  that  the  piraboloid  is  Just  half  of  its  circumscribing 
cylinder. 

(3.)     To  find  the  solidity  of  a  sphere.  ■ 

Let  FC/ revolve  on  the  axis  VC  as  before;  now  on  the  suppo- 
sition that  FC/is  the  arc  of  a  circle  whose  radius  is  /*, 


CALCULUS.  34i> 

Then      (r — x)'^-\-y'^=r^.         y^z=2rx — ar^. 
Then      J'Tti/^dx=ftJ^{2rx^x^)dx=:7i(rx^—^—\-\-c. 

This  integral  requires  no  correction,  because  when  x=0,  y=0 
and  then  the  area  equals  0,  and  c=0. 

This  integral  represents  the  true  value  of  any  segment  corres- 
ponding to  any  assumed  yalue  of  x  between  x=0  and  x=2r. 

If  x=2r  the  segment  will  comprise  the  whole  sphere. 

Then  n{  rx^ — — )==7t{  4r^ — )= 

\  3/        \  3/3 

This  corresponds  to  theorem  17,  book  vii.  Geometry. 
(4.)  The  differential  of is conversely. 

Iniegi'cUe  . Ans. . 

Put  n-{'l=m,    then  n=m — 1,  and  n — l=m — 2. 

With  these  substitutions  the  expression  to  be  integrated  is 

(m—\)x''-^dx    ^^  (m—\)    fx'^^dx(\-^)-'^. 
But  (l+.)-"=l-..+..('!!±i).^-..(-t^)  (^^).3+ 

Multiply  the  second  member  by  x'^'^dx,  then  it  becomes 

-f-  (fee.  ■ 

Now  integrate  each  term  separately,  and  the  result  will  be 

^::ii^x-jL.^x  -^-^f  !!H:i^  V"''-^+ &c.  <fec. 

m—l  '2  2  \     3     /  ' 

Multiply  each  term  by  (m — 1),  observing  that  m — 1  equals  n, 

Thenx''^nx-+n(py-^^-n(^'^^(^^^ 

Replacing  the  value  of  m,  the  series  becomes 


SfiO  ROBINSON'S   SEQUEL. 

.._„....  +«(:«±l)x-  -„(«-ti)  (!±?).-3+ &c. 

By  factoring,  a;"A— 7w;+«'!^«i— fi-^±i.!H::?a;3+  <fec.  ) 


.  Ans, 


(l+x)' 


We  close  this  volume  by  giving  the  two  following  integrations, 
which  troubled  us  very  much  some  years  ago.  They  are  from 
Poisson's  Mecanique  ;  the  first  on  page  223,  vol.  1 ,  the  second  is 
on  page  406  of  the  same  volume. 

Poisson  gives  the  equation 

Then  simply  says,  the  integral  complete  is 

y=c  &m.lxJ—-\-f  V  c  and /being  arbitrary  constants. 

How  did  he  obtain  the  integral? 

Divide  both  members  of  the  equation  by  — &,  and  then  multi- 
ply both  by  2c?y.     Then  we  shall  have 

The  first  member  is  the  differential  of  -r_,  dx^  being  constant. 

dx^ 

The  second  member  is  easily  integrated. 

Then  ^^L.c^—?-y\ 

dx^      h  h^ 

We  add  ( — -c^  J  for  the  arbitrary  constant,  for  (  ._  cM  niay 

represent  any  quantity  as  well  as  c  alone,  and  we  place  it  first, 
because  the  other  term  is  minus.  Taking  the  square  root,  we 
h«v«, 

t-rA'-'-y 


CALCULUS.  351 


2 


Integrating  both  sides,  and 

The  first  member  of  this  equation  is  an  arc  of  a  circle  whose 

sine  is  ^-  and  radius  unity. 
c 

Let  AE  be  that  arc  ;  then 

J)I!z=zl.     DJS=&m,  AH  =sin.  {x^^-{-f\ 
Hence,  ?.=sin.  ^a:^_-+/j 

Integrate     dx — ^anu:dQ=~cos.QdQ. 

This  corressponds  to  the  general  formula, 

di/-\-Pydx=  Qdx, 
Assume  y=zz.ef-^^^  (1) 

Then  theory  gives  the  following  formula  for  the  result : 

y^e-P'^  {  Jef'^  .  Qdx+c)  (F) 

To  apply  this  formula  to  our  equation,  we  must  make 

ic=y,  P=— 2am,  dx^dQ,  ^=:?^cos.  ^,  Fdx  =  —^a7ndQ, 

a 

rFdx=—2amQ, 

Differentiating  (1),  substituting  the  result  in  the  general  f6r- 

mula  and  reducing,  we  find   dz=e^ ^^""^—cos.  QdQ    (2) 

a 

To  integrate  this  last  equation,  we  must   use  the  following 

formula  : 

rudv=uv —  fvdu  (3) 

« It  will  be  a  good  exercise  for  a  learner  to  differentiate  this  equation  twice, 
and  see  if  it  returns  to  the  original. 


3^  ROBINSON'S  SEQUEL. 

a  a 

These  values  put  in  formula  (3),  give 

By  applying  the  same  formula, 

if  we  compare  (2),  (4),  and  (5),  we  shall  perceive  that   the 
ftrst  member  of  (4)  is  z,  and  the  last  term  of  (5)  is  also  z. 
Therefore,  (4)  becomes 

e=e-2-^'<i^^sin.^— e2^™^'^4^mcos.^— 4a2m20 


e 
Or.         0= 


-SaJnQ 


/  _^cos.  Q — ^gm  cos.  Q  \ 


l+4a2m2 
This  value  of  z  put  in  the  general  formula  (■^),  gives 


Let  us  now  reverse  the  operation  and  differentiate  this  equation. 

Thus,  dx=d,{ce-^-^  )+_^^_^.^^?__c?^+.i^«^:^?_cf$ 

To  differentiate  the  first  term  of  the  second  member,  we  put 

«=re2amQ      Then   log.  w=log.  c+2a7W^log.  e 
Observing  that  log.e=l,  and  differentiating  this  last  equation 
we  have  —.^=^amdQ,  ov  du^=^amce'^*^dQ. 


If    «=e-2»«»Q        log.M=— 2amQ        ^==— SamrfQ. 

u 


Whence,  </«=— SamCe-zamQ^/Q, 


CALCULUS.  363 

Whence  ^  =2a7n^^ -"'^4.     g^cos.(g  4^m  sin.  ^ 


Dividing  by  2am, 


._-^^2amQi       2^  sjn.  ^     _   4^m  COS.  ^ 
a(l-l-4a2m2)      (l+4a2m2) 


By  subtraction, 


dx  _         2^cos.  ^       j^  4ffmcos.Q 


-x= 


9,atndQ  2a2m(l+4a2m2)  '  (l+4a2w2) 


That  is     __^_— a;= ^ffcos.Q ,     Ssfa'^m^  cos.Q 

'    2amdQ  2a^m(\^4a^m^y9.a''m(\-\-4a''m'') 

__(l-(-4a^»yi2)2ycos.^_ 2^  cos.Q 

Therefore,       c^ — 2amxdQz=.Rcos.QdQ,  the  original  equation. 

a 


23 


LOGARITHMIC  TABLES; 


ALSO    A   TABLE   OF   THB 


TRIGONOMETRICAL    LINES; 

AND  OTHER  NECESSARY  TABLES. 


' 

LOGARITHMS    OF 

NUMBERS 

FROM 

1      TO      10000* 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0  000000 

26 

1  414973 

51 

1  707570 

76 

1  880814 

2 

0  301030 

27 

1  431364 

52 

1  716003 

77 

1  886491 

3 

0  477121 

28 

1  447158 

53 

1  724276 

78 

1  892095 

4 

0  602060 

29 

1  432398 

64 

1  732394 

79 

1  897627 

5 

0  698970 

30 

1  477121 

65 

1  740363 

80 

1  903090 

6 

0  778151 

31 

1  491362 

56 

1  748188 

81 

1  908485 

7 

0  845098 

32 

1  505150 

57 

1  755875 

82 

1  913814 

8 

0  903090 

33 

1  518514 

58 

1  763428 

83 

1  919078 

9 

0  954243 

34 

1  531479 

69 

1  770852 

84 

1  924279 

10 

1  000000 

35 

1  544068 

60 

1  778151 

85 

1  929419 

11 

1  041393 

36 

1  556303 

61 

1  785330 

86 

1  934498 

12 

1  079181 

37 

1  568202 

62 

1  792392 

87 

1  939519 

13 

1  113943 

38   . 

1  579784 

63 

1  799341 

88 

1  944483 

14 

1  146128 

39 

1  591065 

64 

1  806180 

89 

1  949390 

15 

1  176091 

40 

1  602060 

65 

1  812913 

90 

1  954243 

16 

1  204120 

41 

1  612784 

66 

1  819544 

91 

1  959041 

17 

1  £30449 

42 

1  623249 

67 

1  826075 

92 

1  963788 

18 

1  255273 

43 

1  633468 

68 

1  832509 

93 

1  968483 

19 

1  278754 

44 

1  643453 

69 

1  838849 

94 

1  973128 

20 

1  301030 

45 

1  653213 

70 

1  845098 

95 

1  977724 

21    . 

1  322219 

46 

1  662768 

71    , 

1  851258 

96 

1  982271 

22 

1  342423 

47    • 

1  672098 

72 

1  857333 

97 

1  986772 

23 

1  361728 

48 

1  681241 

73 

1  863323 

98 

1  991226 

24 

1  380211 

49 

1  690196 

74 

1  869232 

99 

1  995635 

25 

1  397940 

50 

1  698970 

75 

1  875661 

100 

2  000000 

N 

.  B.  In  the  following  table,  in  the  last  ni 

ne  columns  of  each  page,  where 

thel 
intrc 
and 
thel 

irst  or  leading  figures  change  from  9's 
>d«ced  instead  of  the  O's  through  the  re 
to  indicate  that  from  thence  the  corre 
irst  column  stands  in  the  next  lower  I 

to  O's,  points  or  dots  are  now 
.st  of  the  line,  to  catch  Ihe  eye, 
spending  natural  numbers  in 
ine,  and  its  annexed  first  two 

figUJ 

'es  «f  the  Logarithms  in  the  second  co 

.ttmn 

LOGARITHMS  OF  NUMBERS.      3 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

100 

000000 

0434 

0868 

1301 

1734 

2166 

2698 

3029 

3461 

3891 

101 

4321 

4750 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

102 

8600 

9026 

9461 

9876 

.300 

.724 

1147 

1570 

1993 

2415 

103 

012837 

3259 

3680: 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

104 

7033 

7461 

7868 

8284 

8700 

9116 

9632 

9947 

.361 

.775 

105 

021189 

1603 

2016 

3428 

2841 

3252 

3664 

4075 

4486 

4896 

108 

5306 

5715 

6125 

6633 

6942 

7350 

7767 

8164 

8671 

8978 

107 

9384 

9789 

.195 

.600 

1004 

1408 

1812 

2216 

2619 

3021 

108 

033424 

3826 

4227 

4628 

502» 

5430 

5830 

6230 

6629 

7028 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

110 

041393 

1787 

2182 

2576 

2969 

3362 

3765 

4148 

4540 

4932 

111 

5323 

5714 

6106 

6496 

6886 

7276 

7664 

8053 

8442 

8830 

112 

9218 

9606 

9993 

.380 

.766 

1163 

1638 

1924 

2309 

2694 

113 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

114 

6905 

7286 
1075 

7666" 

8046 

8426 

8805 

9186 

9563 

9942 

.320 

115 

060698 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

116 

4458 

4832 

6208 

5580 

6963 

6326 

6699 

7071 

7443 

7815 

117 

•  sise 

8557 

8328 

92S8 

S6C8 

..38 

.407 

.776 

1146 

1614 

118 

071882 

2250 

2617 

2985 

3362 

3718 

4086 

4451 

4816 

6182 

119 

6647 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

120 

9181 

9543 

9904 

.266 

.626 

.987 

1347 

1707 

2067 

2426 

121 

082V85 

3144 

3503 

3861 

4219 

4676 

4934 

5291 

5647 

6004 

122 

6360 

6716. 

7071 

7426 

7781 

8136 

8490 

8846 

9198 

9562 

123 

9905 

.268 

.611 

.963 

1315 

1667 

2018 

2370 

2721 

3071 

124 

093422 

3772 

4122 

4471 

4820 

5169 

5618 

5866 

6216 

6662 

125 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

0026 

126 

100371 

071& 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

127 

3804 

4146 

4487 

4828 

6169 

5510 

5851 

6191 

6631 

6871 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9679 

9916 

.263 

129 

110590 

0926 

1263 

1599 

1934 

2270 

2606 

2940 

3275 

3609 

130 

3943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

131 

7271 

7603 

7934 

8265 

8596 

8926 

9256 

9586 

9915 

0246 

132 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

133 

3852 

4178 

4604 

4830 

6166 

6481 

5806 

6131 

6466 

6781 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

..12 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

136 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

137 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

138 

9879 

.194 

.508 

.822 

1136 

1450 

1763 

2076 

2389 

2702 

139 

143015 

3327 

3639 

3951 

4263 

4574 

4886 

5196 

6607 

6818 

140 

6128 

6438 

6748 

7068 

7367 

7676 

7985 

.8294 

8603 

8911 

141 

9219 

9527 

9836 

.142 

.449 

,756 

1063 

1370 

1676 

1982 

142 

152288 

2594 

2900 

3206 

2610 

3815 

4120 

4424 

4728 

6032 

143 

5336 

5640 

5943 

6246 

6549 

6852 

7164 

7457 

7769 

8061 

144 

8362 

8664 

8965 

9266 

9567 

9868 

.168 

.469 

.769 

1068 

145 

161368 

1667 

1967 

2266 

2664 

2863 

3161 

3460 

3758 

4055 

146 

4353 

4650 

4947 

5244 

5641 

5838 

6134 

6430 

6726 

7022 

147 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

148 

170262 

0565 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

6612 

6802 

4 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

150 

176091 

6381 

6670 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

151 

8977 

9264 

9552 

9839 

.126 

.413 

.699 

.985 

1272 

1568 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3565 

3839 

4123 

4407 

153 

4691 

4975 

5259 

5642 

5825 

6108 

6391 

6674 

6956 

7239 

164 

7621 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

..51 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

156 

3125 

3403 

3681 

3959 

4237 

4614 

4792 

5069 

5346 

6623 

157 

5899 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

158 

8657 

8932 

9206 

9481 

9765 

..29 

.303 

.677 

.860 

1124 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3677 

3848 

160 

4120 

4391 

4663 

4934 

5204 

5476 

5746 

6016 

6286 

6566 

161 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

162 

9515 

9783 

..51 

.319 

.586 

.853 

1121 

1388 

1654 

1921 

163 

212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4679  . 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

165 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

166 

220108 

0370 

0631 

0892 

1163 

1414 

1676 

1936 

2196 

2456 

167 

2716 

29/6 

3236 

3496 

3765 

4016 

4274 

4633 

4792 

5051 

168 

5309 

5568 

5S26 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

169 

7887 

8144 

8400 

8667 

8913 

9170 

9426 

9682 

9938 

.193 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

171 

2996 

3250 

8504 

3767 

4011 

4264 

4517 

4770 

6023 

5276 

172 

5528 

5781 

6033 

6286 

6537 

6789 

7041 

7292 

7544 

7795 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

..50 

.300 

174 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

175 

3038 

3283 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

176 

5513 

5759 

6006 

6262 

6499 

6746 

6991 

7237 

7482 

7728 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

.176 

178 

250420 

0664 

0908 

1151 

1396 

1638 

1881 

2125 

2368 

2610 

179 

2863 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

181 

7679 

7918 

8168 

8398 

8637 

8877 

9116 

9356 

9594 

9833 

182 

260071 

0310 

0548 

0787 

1026 

1263 

1601 

1739 

1976 

2214 

183 

2451 

2688 

2926 

3162 

3399 

3686 

3873 

4109 

4346 

4582 

184 

4818 

5054 

6290 

6525 

5761 

5996 

6232 

6467 

6702 

6937 

185 

7172 

7406 

7641 

7875 

8110 

8344 

8678 

8812 

9046 

9279 

186 

9513 

9746 

9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

187 

271842 

2074 

2306 

2638 

2770 

3001 

3233 

3464 

3696 

3927 

188 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

189 

6462 

6692 

6921 

7161 

7380 

7609 

7838 

8067 

8296 

8526 

190 

8754 

8982 

9211 

9439 

9667 

9895 

.123 

.351 

.578 

.806 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

192 

3301 

3527 

3768 

3979 

A205 

4431 

4656 

4882 

5107 

5332 

193 

5557 

5782 

6007 

6232 

6466 

6681 

6905 

7130 

7354 

7578 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

195 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

196 

2258 

2478 

2699 

2920 

3141 

3363 

3684 

3804 

4026 

4246 

197 

4466 

4687 

4907 

6127 

5347 

5667 

6787 

6007 

6226 

6446 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8636 

199 

8853 

9071 

9289 

9507 

9725 

9943 

.161 

.378 

.596 

.813 

OF  NUMBERS               5 

N. 

0 

1 

2 

3 

4 

g 

6 

7 

8 

9 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

202 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6864 

7068 

7282 

203 

7496 

7710 

7924 

8137 

8361 

S564 

8778 

8991 

9204 

9417 

204 

9630 

9843 

..56 

.268 

.481 

.693 

.906 

1118 

1330 

1542 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130- 

5340 

5661 

5760 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

209 

320146 

0354 

0562 

0769 

0977 

1184 

1391 

1698 

1805 

2012 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

211 

4282 

4488 

4694 

4899 

6105 

5310 

5516 

5721 

5926 

6131 

212 

6336 

6541 

6745 

6950 

7155 

7359 

7663 

7767 

7972 

8176 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

...8 

.211 

214 

330414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

216 

4454 

4655 

4866 

5057 

5257 

5458 

6668 

6859 

6059 

6260 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

218 

8456 

8656 

8865 

9054 

9253 

9451 

9660 

9849 

..47 

.246 

219 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

221 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

6766 

5962 

6167 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7625 

7720 

7915 

8110 

223 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

..54 

224 

350248 

0442 

0636 

^829 

1«23 

1216 

1410 

1603 

1796 

1989 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

226 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7654 

7744 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9466 

9646 

229 

9835 

..25 

.215 

.404 

.593 

.783 

.972 

1161 

1350 

1539 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

231 

3612 

3800 

3988 

4176 

4363 

4561 

4739 

4926 

5113 

5301 

232 

5488 

5676 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

234 

9216 

9401 

9587 

9772 

9968 

.143 

.328 

.513 

.698 

.883 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4016 

4198 

4382 

4565 

237 

4748 

4932 

6115 

5298 

5481 

6664 

5846 

6029 

6212 

6394 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

239 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

..30 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

241 

2017 

2197 

2377 

2657 

2737 

2917 

3097 

3277 

3456 

3636 

242 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

6249  1 

5428 

243 

5606 

5785 

5964 

6142 

63^1 

6499 

6677 

6856 

7034 

7212 

244 

7390 

7568 

7746 

7923 

8101 

8279 

8466 

8634 

8811 

8989 

245 

9166 

9343 

9520 

9698 

9875 

..51 

.228 

.405 

.582 

.759 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

247 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

248 

4452 

4627 

4802 

4977 

5152 

5826 

5601 

6676 

5850  ! 

6025 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592  j 

7766 

6 

LOGARITHMS 

N. 

0 

1 

2     3 

4 

5 

6 

7 

8 

9 

250 

397940 

8114 

8287 

8461 

8634  8808  | 

8981 

9154 

9328 

9501 

261 

9674 

9847 

..20 

,192 

.366 

,538 

.711 

.883 

1056 

1228 

252 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

253 

3121 

3292 

3464 

8636 

3807 

3978 

4149 

4320 

4492 

4663 

254 

4834 

5006 

5176 

6346 

5517 

5688 

5858 

6029 

6199 

6370 

255 

6540 

6710 

6881 

7061 

7221 

7391 

7561 

7731 

7901 

8070 

266 

8240 

8410 

8679 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

267 

9933 

.102 

.271 

.440 

.609 

.777 

.946 

1114 

1283 

1451 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

250 

3300 

3467 

3635 

3803 

8970 

4137 

4305 

4472 

4639 

4806 

260 

4973 

5140 

5307 

5474 

5641 

6808 

5974 

6141 

6308 

6474 

261 

6641 

6807 

6973 

7189 

7308 

7472 

7638 

7804 

7970 

8135 

262 

8301 

H467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

263 

9956 

.121 

.286 

.451 

.616 

,781 

.946 

1110 

1275 

1439 

264 

421604 

1788 

1933 

2097 

2261 

2426 

2590 

2764 

2918 

3082 

265 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

266 

4882 

5045 

5208 

f-371 

C634 

5697 

5860 

6023 

6186 

6349 

267 

6511 

6874 

G83o 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

269 

9762 

9914 

..75 

;  .236 

.398 

.559 

.720 

.881 

1042 

1203 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

272 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

273 

6163 

6322 

6481 

6640 

6800 

6957 

7116 

7275 

7433 

7592 

274 

7751 

7909 

8067 

8226 

,8384 

8542 

8701 

8859 

9017 

9175 

5i75 

9333 

9491 

9648 

9805 

9964 

.122 

.279 

.437 

.594 

.752 

276 

440909 

1066 

1224 

1381 

1538 

1695 

1862 

2009 

2166 

2323 

277 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3676 

3732 

3889 

278 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

279 

5604. 

6760 

5915 

6071 

6226 

6382 

6637 

6692 

6848 

7003 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8897 

8552 

281 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

..95 

282 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

283 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

284 

3318 

3471 

3624 

3777 

3930 

4082 

4286 

4387 

4540 

4693 

285 

4845 

4997 

5150 

5302 

5454 

5603 

6758 

5910 

6062 

6214 

286 

6366 

6618 

6670 

6821 

6973 

7125 

7276 

7428 

7679 

7731 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8-940 

9091 

9242 

288 

9392 

9543 

9694 

9845 

9995 

.146 

.296 

.447 

.597 

.748 

289 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

292 

5383 

5532 

5680 

6829 

5977 

6126 

6274 

6423 

6671 

6719 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9283 

9380 

9627 

9676 

295 

9822 

9969 

.116 

.263 

.410 

.567 

.704 

.851 

.998 

1145 

296 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

£464 

2610 

297 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

8779 

8925 

4071 

298 

4216 

4362 

4508 

4(>53 

4799 

4944 

5090 

5235 

5381 

6526 

299 

5G71 

6816 

6962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

OF  NUMBERS.              7 

N. 

0 

I 

2 

3 

4 

6 

6 

7 

8 

9 

300 

477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

8278 

8422 

301 

8566 

8711 

8855 

8999 

9143 

9287 

9481 

9575 

9719 

9863 

302 

480i)07 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156 

1299 

303 

1443 

1686 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

304 

2874 

3016 

3159 

3302 

3445 

3587 

8730 

3872 

4015 

4167 

305 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

6295 

5437 

5579 

306 

6721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6856 

6997 

307 

7138 

7280 

7421 

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8127 

8269 

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308 

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9667 

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309 

9959 

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.239 

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.661 

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.941 

1081 

1222 

310 

491362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

311 

2760 

2900 

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3319 

3458 

3697 

3737 

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4015 

312 

4163 

4294 

4433 

4572 

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4850 

4989 

5128 

5267 

6406 

313 

6544 

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5822 

5960 

6099 

6238 

6376 

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6791 

314 

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7897 

8035 

8173 

315 

8311 

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9137 

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9412 

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316 

9687 

9824 

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317 

601059 

1196 

1333 

1470 

1607 

1744 

1880 

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2154 

2291 

318 

2427 

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2700 

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2973 

3109 

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3382 

3518 

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319 

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320 

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5283 

5421 

5557 

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5828 

5964 

6093 

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6370 

321 

6505 

6640 

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6911 

7046 

7181 

7316 

7451 

7583 

7721 

322 

7856 

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8126 

8260 

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8934 

9008 

323 

9203 

9337 

9471 

9606 

9740 

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.143 

.277 

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324 

510546 

0679 

0813 

0947 

1081 

1215 

1349 

1482 

1616 

1750 

325 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3034 

326 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4414 

327 

4548 

4681 

4813 

4946 

6079 

5211 

5344 

5476 

5609 

5741 

328 

5874 

6006 

6139 

6271 

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6668 

6800 

6932 

7064 

329 

7196 

7328 

7460 

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7987 

8119 

8251 

8382 

330 

8514 

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9040 

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9434 

9566 

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331 

9828 

9959 

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.221 

.353 

.484 

.615 

.745 

.876 

1007 

332 

521138 

1269 

1400 

1630 

1661 

1792 

1922 

2053 

2183 

2314 

333 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

334 

3746 

3876 

4006 

4136 

4266 

4396 

4626 

4656 

4785 

4916 

335 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

336 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

337 

7630 

7759 

7888 

8016 

8146 

8274 

8402 

8531 

8660 

8788 

338 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

..72 

339 

530200 

0328 

0466 

0584 

0712 

0840 

0968 

1096 

1223 

1L61 

340 

1479 

1607 

1734 

1862 

1960 

2117 

2245 

2372 

2500 

2f27 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

342 

4026 

4163 

4280 

4407 

4534 

4661 

4787 

4914 

6041 

5167 

343 

6294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

344 

6658 

6685 

6811 

6937 

7060 

7189 

7316 

7441 

7667 

7693 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

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346 

9076 

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9452 

9678 

9703 

9829 

9954 

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347 

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0465 

0580 

0705 

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0956 

1080 

1205 

1330 

1454 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2462 

2576 

2701 

349 

2825 

2960 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

8 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

350 

544068 

4192 

4316 

4440 

4664 

4688 

4812 

4936 

5060 

5183 

351 

5307 

5431 

5555 

5678 

5805 

5925 

6049 

6172 

6296 

6419 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

363 

7775 

7898 

8021 

8144 

8267 

8389 

8612 

8636 

8758 

8881 

354 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

.196 

355 

550228 

0351 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

357 

2668 

2790 

2911 

3033 

3155 

8276 

3393 

3519 

8640 

3762 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

359 

5094 

6216 

5346 

5467 

5578 

5699 

5820 

5940 

6061 

6182 

360 

6303 

6423 

6544 

6664 

6786 

6905 

7026 

7146 

7267 

7387 

361 

7507 

7G27 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

362 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

863 

9907 

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.146 

.266 

.886 

.504 

.624 

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.863 

.982 

364 

561101 

1221 

1340 

1459 

1678 

1698 

1817 

1986 

2055 

2173 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

366 

8481 

3600 

3718 

3837 

3956 

4074 

4192 

4311 

4429 

4548 

367 

4666 

4784 

4903 

5021 

5139 

5267 

6376 

5494 

5612 

5780 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

369 

7026 

7144 

7262 

7879 

7497 

7614 

7732 

7849 

7967 

8084 

370 

8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

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371 

9374 

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9608 

9725 

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9959 

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372 

570543 

0660 

0776 

0898 

1010 

1126 

1243 

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1476 

1692 

373 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

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2639 

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374 

2872 

2988 

3104 

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375 

4031 

4147 

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4494 

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4957 

5072 

376 

5188 

5303 

6419 

5634 

5660 

5766 

5880 

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6111 

6226 

377 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

378 

7492 

7607 

7722 

7836 

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8066 

8181 

8296 

8410 

8625 

379 

8639 

8764 

8868 

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9212 

9826 

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9555 

9669 

880 

9784 

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.811 

381 

580926 

1039 

1163 

1267 

1381 

1496 

1608 

1722 

1836 

1950 

382 

2063 

2177 

2291 

2404 

2618 

2631 

2746 

2868 

2972 

8085 

383 

3199 

3312 

3426 

8639 

3652 

3766 

3879 

3992 

4105 

4218 

384 

4331 

4444 

4667 

4670 

4783 

4896 

5009 

5122 

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5348 

385 

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5912 

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6137 

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6700 

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7149 

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7823 

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8160 

8272 

8384 

8496 

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8832 

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9615 

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389 

9950 

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.953 

390 

591065 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1965 

2066 

391 

2177 

2288 

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2510 

2621 

2732 

2843 

2964 

8064 

3175 

392 

3286 

3397 

8508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

393 

4393 

4603 

4614 

4724 

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4945 

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5165 

5276 

5386 

394 

5496 

5606 

5717 

5827 

6937 

6047 

6157 

6267 

6877 

6487 

395 

6507 

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6817 

6927 

7037 

7146 

7256 

7866 

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7586 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

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397 

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398 

9883 

9992 

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.537 

.646 

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399 

600973 

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1191 

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1408 

1617 

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1734 

1843 

1951 

OF  NUMBERS.              9 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

400 

602060 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

401 

3144 

3253 

3361 

3469 

3573 

3686 

3794 

3902 

4010 

4118 

402 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

403 

5305 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

404 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

405 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

406 

8526 

8633 

8740 

8847 

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9061 

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9381 

9488 

407 

9594 

9701 

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9914 

..21 

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408 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

409 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

411 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

412 

4897 

5003 

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5213 

5319 

5424 

5529 

5634 

5740 

5846 

413 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

6895 

414 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7754 

7839 

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415 

8048 

8153 

8257 

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8780 

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416 

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417 

620136 

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0552 

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0760 

0864 

0068 

1072 

418 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

6107 

5210 

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5312 

5415 

5518 

5621 

6724 

5827 

5929 

6032 

6135 

6238 

423 

6340 

6443 

6546 

6648 

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6956 

7058 

7161 

7263 

424 

7366 

7468 

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425 

8389 

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427 

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1139 

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1647 

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1951 

2052 

2153 

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2356 

429 

2457 

2559 

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2761 

2862 

2963 

3064 

3165 

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3468 

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3973 

4074 

4175 

4276 

4376 

431 

4477 

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4679 

4779 

4880 

4981 

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5182 

5283 

5383 

432 

5484 

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6685 

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6187 

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433 

6488 

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6789 

6889 

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7089 

7189 

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434 

7490 

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435 

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437 

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1177 

1276 

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438 

1474 

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1672 

1771 

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1970 

2069 

2168 

2267 

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439 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

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3354 

440 

3453 

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4340 

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6226 

5324 

442 

5422 

5521 

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6011 

6110 

6208 

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443 

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6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7286 

444 

7383 

7481 

7579 

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7774 

7872 

7969 

8067 

8165 

8262 

445 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

446 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

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.113 

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447 

650308 

0405 

0502 

0599 

0696 

0793 

0890 

0987 

1084 

1181 

448 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2160 

449 

2246 

2343 

2440 

2530 

2633 

2730 

2826 

2923 

3019 

3116 
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10 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

450 

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3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

461 

4177 

4273 

4369 

4466 

4562 

4658 

4754 

4850 

4946 

6042 

453 

5138 

6235 

5331 

5427 

5626 

6619 

6715 

5810 

5906 

6002 

453 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

454 

7056 

7162 

7247 

7343 

7438 

7634 

7629 

7725 

7820 

7916 

456 

8911 

8107 

8202 

8298 

8398 

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8584 

8679 

8774 

8870 

456 

8966 

9060 

9156 

9250 

9346 

9441 

9536 

9631 

9726 

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457 

9916 

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.296 

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.676 

.771 

458 

660866 

0960 

1055 

1150 

1245 

1339 

1434 

1629 

1623 

1718 

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1813 

1907 

2002 

2096 

2191 

2286 

2380 

2475 

2569 

2663 

460 

2768 

2852 

2947 

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3135 

3230 

3324 

3418 

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3701 

3796 

3889 

3983 

4078 

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4648 

462 

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6018 

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67d9 

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6424 

464 

6518 

6612 

6705 

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7079 

7173 

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7360 

465 

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466 

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1173 

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1728 

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2005 

470. 

2098 

2190 

2283 

2375 

2467 

2560 

2652 

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3297 

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479 

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1422 

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1603 

1693 

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1874 

1964 

2056 

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2145 

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2777 

2867 

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3047 

3137 

3227 

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7083 

7172 

7261 

7351 

7440 

487 

7629 

7618 

7707 

7796 

7886 

7976 

8064 

8153 

8242 

8331 

488 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

489 

9309 

9398 

9486 

9576 

9664 

9753 

9841 

9930 

..19 

.107 

490 

690196 

0285 

0373 

0362 

0550 

0639 

0728 

0816 

0905 

0993 

491 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

492 

1966 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

493 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

494 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

495 

4605 

4693 

4781 

4868 

4956 

6044 

5131 

5210 

5307 

5394 

496 

5482 

6669 

6657 

6744 

5832 

6919 

6007 

6094 

6182 

6269 

497 

6356 

5444 

6531 

6618 

6706 

6793 

6880 

6968 

7056 

7142 

498 

7229 

7317 

7404 

7491 

7678 

7666 

7752 

7889 

7926 

8014 

499 

8101 

8188 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

b«83 

OF  NUMBERS.             11 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

500 

693970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

501 

9838 

9924 

..11 

..98 

.184 

.271 

.358 

.444 

.631 

.617 

502 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1396 

1482 

503 

1568 

1664 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

504 

2431 

2617 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3206 

505 

3291 

3377 

3463 

3549 

8636 

3721 

3807 

3896 

3979 

4065 

508 

4151 

4236 

4322 

4408 

4494 

4579 

4666 

4751 

4837 

4922 

507 

5008 

5094 

5179 

5266 

6350 

6436 

6522 

6607 

5693 

5778 

508 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6647 

6632 

509 

6718 

6803 

6888 

6974 

7069 

7144 

7229 

7315 

7400 

7485 

610 

7570 

7655 

7740 

7826 

7910 

7996 

8081 

8166 

8261 

8336 

511 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9186 

512 

9270 

9356 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

..33 

513 

710117 

0202 

6287 

0371 

0466 

0540 

0625 

0710 

0794 

0879 

514 

0963 

1048 

1132 

1217 

1301 

1386 

1470 

1554 

1639 

1723 

515' 

1807 

1892 

1976 

2660 

2144 

2229 

2313 

2397 

2481 

2566 

516 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3326 

3407 

517 

3491 

5576 

3659 

3742 

3826 

3910 

8994 

4078 

4162 

4246 

518 

4330 

4414 

4497 

4581 

4666 

4749 

4833 

4916 

5000 

5084 

519 

5167 

5251 

6336 

5418 

5602 

6586 

5669 

5763 

5836 

6920 

520 

6003 

6087 

6170 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

521 

6838 

6921 

7004 

7688 

7171 

7254 

7338 

7421 

7504 

7587 

522 

7671 

7764 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

523 

8502 

8585 

8668 

8761 

8834 

8917 

9000 

9083 

9165 

9248 

524 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

..77 

525 

720159 

0242 

0326 

0407 

0490 

6573 

0655 

0738 

0821 

0903 

526 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1663 

1646 

1728 

527 

1811 

1893 

.975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

528 

2634 

3716 

2798 

2881 

2963 

3046 

3127 

3209 

8291 

3374 

529 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

630 

4276 

4358 

4440 

4622 

4604 

4686 

4767 

4849 

4931 

5013 

531 

5096 

5176 

5268 

5340 

6422 

5503 

6585 

5667 

5748 

5830 

532 

5912 

6993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

533 

6727 

6809 

6890 

6972 

7653 

7134 

7216 

7297 

7379 

7460 

534 

7641 

7623 

7704 

7786 

7866 

7948 

8029 

8110 

8191 

8273 

535 

8354 

8435 

8516 

8597 

867S 

8759 

8841 

8922 

9003 

9084 

536 

9166 

9246 

9327 

9403 

9489 

9570 

9651 

9732 

9813 

9893 

537 

9974 

..65 

.136 

.217 

.298 

.378 

.469 

.440 

.621 

.702 

538 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

539 

1689 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

540 

2394 

2474 

2656 

2636 

2716 

2796 

2876 

2966 

3037 

3117 

541 

3197 

3278 

3368 

3438 

3518 

3598 

3679 

3769 

3839 

3919 

542 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

543 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

5619 

544 

6399 

6679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

545 

6397 

6476 

6656 

6636 

6715 

6796 

6874 

6954 

7034 

7113 

546 

7193 

7272 

7352 

7431 

7611 

7590 

7670 

7749 

7829 

7908 

547 

7987 

8067 

8146 

8226 

8305 

8384 

8463 

8543 

8622 

8701 

548 

8781 

8860 

8939 

9018 

9097 

9177 

9266 

9335 

9414 

•9493 

549 

9672 

9651 

9731 

9810 

9889 

9968 

..47 

.126 

.205 

.284 

12 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

650 

740363 

0442 

0521 

0560 

0678 

0757 

0836 

0915 

0994 

1073 

551 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

552 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2646 

553 

2726 

28(M 

2882 

2961 

3039 

3118 

3196 

3276 

3353 

3431 

554 

3510 

3568 

3667 

3745 

3823 

3902 

8980 

4058 

4136 

4215 

555 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

556 

5075 

5163 

5231 

5309 

5387 

6465 

5543 

6621 

6699 

6777 

657 

5855 

6933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6656 

558 

6634 

6712 

6790 

6868 

6946 

7023 

7101 

7179 

7256 

7334 

559 

7412 

7489 

7567 

7646 

7722 

7800 

7878 

7956 

8033 

8110 

560 

8188 

8266 

8343 

8421 

8498 

8676 

8653 

8731 

8808 

8885 

561 

8963 

9040 

9118 

9196 

9272 

9360 

9427 

9504 

9582 

9659 

562 

9736 

9814 

9891 

9968 

..45 

.123 

.200 

.277 

.354 

.431 

663 

750508 

0586 

0663 

0740 

0817 

0894 

0971 

1048 

1126 

1202 

564 

1279 

1356 

1433 

1610 

1587 

1664 

1741 

1818 

1895 

1972 

565 

2048 

2125 

2202 

2279 

2356 

2433 

2609 

2586 

2663 

2740 

566 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3606 

567 

3582 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4196 

4272 

568 

4348 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

5036 

569 

5112 

5189 

5266 

6341 

5417 

6494 

5670 

6646 

5722 

6799 

570 

5875 

6951 

6027 

6103 

6180 

6256 

6332 

6408 

6484 

6560 

571 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

572 

7396 

7472 

7648 

7624 

7700 

7775 

7851 

7927 

8003 

8079 

573 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

574 

8912 

8988 

9068 

9139 

9214 

9290 

9366 

9441 

9517 

9692 

575 

9668 

9743 

9819 

9894 

9970 

..45 

.121 

.196 

.272 

.347 

576 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0060 

1025 

1101 

&77 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

578 

1938 

2003 

2078 

2163 

2228 

2303 

2378 

2453 

2529 

2604 

579 

2679 

2754 

2829 

2904 

2978 

3053 

3128 

2203 

3278 

3353 

580 

34S8 

3503 

3578 

3653 

3727 

3802 

3877 

3962 

4027 

4101 

581 

4176 

4261 

4326 

4400 

4476 

4650 

4624 

4699 

4774 

4848 

582 

4923 

4998 

5072 

5147 

5221 

5296 

6370 

5445 

6520 

6594 

583 

6669 

5743 

5818 

6892 

6956 

6041 

6115 

6190 

6264 

6338 

584 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

585 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

586 

T.89S 

7972 

80i6 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

687 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9166 

9230 

9303 

588 

9377 

9461 

9525 

9699 

9673 

9746 

9820 

9894 

9968 

..42 

689 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

591 

1687 

1661 

1734 

1808 

1881 

1956 

2028 

2102 

2175 

2248 

592 

2322 

2395 

2468 

3542 

2615 

2688 

2762 

2836 

2908 

2981 

593 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

594 

3786 

3860 

3933 

4006 

40?9 

4152 

4225 

4298 

4371 

4444 

595 

4617 

4590 

4663 

4736 

4809 

4882 

4966 

5028 

5100) 

6173 

596 

5246 

5319 

5392 

5465 

5538 

6610 

5683 

5756 

5829 

5902 

597 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

698 

6701 

6774 

6846 

6919 

0992 

7064 

7137 

7209 

7282 

7354 

699 

7427 

7409 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

^ 

OF  NUMBERS.              13 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

600 

778151 

8224 

8296 

8368 

8441 

8513 

8585 

8658 

8730 

8802 

601 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

602 

9596 

6669 

9741 

9813 

9885 

9967 

..29 

.101 

.173 

.245 

603 

780317 

0389 

0461 

0533 

0505 

0677 

0749 

0821 

0893 

0965 

604 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

605 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

606 

2473 

2644 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

607 

3189 

3260 

3332 

3403 

8476 

3646 

3618 

8689 

3761 

3832 

608 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

609 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

6615 

5686 

5757 

5828 

5899 

5970 

611 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

612 

6761 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

613 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

614 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

615 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

616 

9581 

9651 

9722 

9792 

9863 

9933 

...4 

..74 

.144 

.216 

617 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

618 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

619 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

8022 

621 

8092 

3162 

3231 

3301 

8371 

3441 

3511 

3581 

8661 

3721 

622 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4849 

4418 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

624 

5186 

5254 

5324 

5393 

6463 

5532 

5602 

5672 

5741 

5811 

626 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

626 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

627 

7268 

7337 

7406 

7476 

7545 

7614 

7683 

7752 

7821 

7890 

628 

7960 

8029 

8098 

8167 

8236 

8305 

8874 

8443 

8513 

8582 

629 

8661 

8720 

8789 

8858 

8927 

8996 

9066 

6134 

9203 

9272 

630 

9341 

9409 

9478 

9547 

9610 

9685 

9754 

9823 

9892 

9961 

631 

800026 

0098 

0167 

0236 

0305 

0373 

0442 

0511 

0580 

0848 

632 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

633 

1404 

1472 

1541 

1609 

1678 

1747 

1816 

1884 

1952 

2021 

634 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2668 

2637 

2705 

635 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

8252 

3321 

3389 

636 

3457 

3526 

3594 

3662 

3730 

3798 

8867 

3935 

4003 

4071 

637 

4139 

4208 

4276 

4354 

4412 

4480 

4548 

4616 

4685 

4753 

638 

4821 

4889 

4957 

5025 

6093 

6161 

5229 

5297 

5365 

5433  1 

639 

5601 

5669 

5637 

6706 

6773 

6841 

5908 

5976 

6044 

6112  j 

640 

6180 

6248 

6316 

6384 

6451 

6519 

6587 

6655 

6723 

6790  ! 

641 

6858 

6926 

6994 

7061 

7129 

7157 

7264 

7332 

7400 

7467   ! 

642 

7635 

7603 

7670 

7738 

7808 

7873 

7941 

8008 

8076 

8143   1 

643 

8211 

8279 

8346 

8414 

8481 

8549 

8616 

8684 

8751 

8818  i 

644 

8886 

8963 

9021 

9088 

9166 

9223 

9290 

9358 

9426 

9492 

645 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

..31 

..98 

.165 

646 

810238 

0300 

0367 

0434 

0501 

0596 

0636 

0703 

0770 

0837 

647 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

648 

1676 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

649 

2246 

2312 

2379 

2446 

2512 

2o79 

2648 

2713 

2780 

2847 

14 

LOGARITHMS 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

6B0 

812913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

651 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

652 

4248 

4314 

4381 

4447 

4514 

4681 

4647 

4714 

4780 

4847 

663 

4913 

4980 

5046 

6113 

5179 

5246 

5312 

5378 

5445 

5511 

654 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

665 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

656 

6904 

6970 

7036 

7102 

7169 

7233 

7301 

7367 

7433 

7499 

657 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

658 

8226 

8292 

8368 

8424 

8490 

8556 

8622 

8688 

8764 

8820 

659 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

660 

9544 

9610 

9676 

9741 

9807 

9873 

9939 

...4 

..70 

.136 

661 

820201 

0267 

0333 

0399 

0464 

0530 

0695 

0661 

0727 

0792 

662 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

663 

1514 

1579 

1646 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

664 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2766 

666 

2822 

2887 

2962 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

666 

3474 

3539 

3605 

3670 

3736 

3800 

3865 

3930 

3996 

4061 

667 

4126 

4191 

4266 

4321 

4386 

4451 

4516 

4681 

4646 

4711 

668 

4776 

4841 

4906 

4971 

6036 

5101 

6166 

5231 

5296 

5361 

669 

5426 

5491 

6656 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

671 

6723 

6787 

6862 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

672 

7369 

7434 

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7628 

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673 

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8080 

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8467 

8531 

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674 

8660 

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9046 

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677 

830589 

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0717 

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0973 

1037 

1102 

1166 

678 

1230 

1294 

1358 

1422 

1486 

1560 

1614 

1678 

1742 

1806 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

680 

2509 

2673 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

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681 

3147 

3211 

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3402 

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4039 

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4166 

4230 

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4367 

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4421 

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5056 

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5183 

6247 

5310 

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5437 

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6564 

5627 

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5691 

6754 

5817 

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5944 

6007 

6071 

6134 

6197 

6261 

686 

6324 

6387 

6461 

6614 

6577 

6641 

6704 

6767 

6830 

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6957 

7020 

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8912  8975 

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0294 

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1046 

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1234 

1297 

694 

1359 

1422  1  1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

695 

1985 

2047  2110 

2172 

2235 

2297 

2360 

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2484 

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2609 

2672  2734 

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2869 

2921 

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3108 

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3233 

3295  3357 

3420 

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3606 

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3918  3980 

4042 

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4166 

4229 

4291 

4363 

4416 

699 

4477 

4539  4601 

4664 

4726 

4788 

4850 

4912 

4974 

6036 

OF  NUMBERS.             15 

N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

j  700 

845098 

5160 

6222 

5284 

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6666 

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6904 

5966 

6028 

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6151 

6213 

6'276 

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6337 

6399 

6461 

6623 

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6646 

6708 

6770 

6832 

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703 

6955 

7017 

7079 

7141 

7202 

7264 

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704 

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7634 

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8004 

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1  705 

8189 

8251 

8312 

8374 

8435 

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8559 

8620 

8682 

8743 

1  706 

8805 

8866 

8928 

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9051 

9112 

9174 

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9297 

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707 

9419 

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9542 

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9665 

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9849 

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9972 

703 

850033 

0095 

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0340 

0401 

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0769 

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0952 

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1136 

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710 

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1503 

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1747 

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1870 

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1992 

2053 

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2236 

2297 

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3394 

3455 

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3577 

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714 

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4002 

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715 

4306 

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1339 

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1616 

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1673 

1631 

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2040 

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16 

LOGARITHMS 

N. 

0 

1 

2 

3     4 

6 

6 

7 

8 

9 

750 

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5119 

6177 

6235 

6293 

6351 

6409 

5466 

5524 

5582 

751 

6640 

5698 

6756 

5813 

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6929 

5987 

6045 

6102 

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762 

6218 

6276 

6333 

6391 

6449 

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6564 

6622 

6680 

6737 

763 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

764 

7371 

7429 

7487 

7644 

7602 

7659 

7717 

7774 

7832 

7889 

756 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

766 

8522 

8579 

8637 

8694 

8752 

8809 

886(5 

8924 

8931 

9039 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

758 

9669 

9726 

9784 

9841 

9898 

9956 

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769 

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0299 

0356 

0413 

0471 

0528 

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0928 

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1042 

1099 

1156 

1213 

1271 

1328 

761 

1385 

14^42 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

762 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

763 

2525 

2581 

2638 

2695 

2752 

2809 

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2923 

2980 

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3150 

3207 

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3377 

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765 

3661 

3718 

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3832 

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4002 

4059 

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4172 

766 

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4285 

4342 

4399 

4455 

4512 

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4909 

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5022 

5078 

6135 

5192 

5248 

6305 

768 

5361 

5418 

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5813 

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6039 

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6209 

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6321 

6378 

6434 

770 

6491 

6547 

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6660 

6716 

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771 

7054 

7111 

7167 

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7280 

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7392 

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7506 

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7617 

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773 

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774 

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777 

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1314 

1370 

1426 

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779 

1537 

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1649 

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1760 

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2039 

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6030 

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6140 

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6416 

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7132 

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7362 

7407 

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7617 

7672 

790 

7627 

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7737 

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7847 

7902 

7957 

8012 

8067 

8122 

791 

8176 

8231 

8286 

8341 

8396 

8451 

8606 

8561 

8615 

8670 

792 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

793 

9273 

9328 

9383 

9437 

9492 

9647 

9602 

9656 

9711 

9766 

794 

9821 

9875 

9930 

9985 

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.149 

.203 

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795 

900367 

0422 

0476 

0531 

0586 

0640 

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0749 

0804 

0859 

796 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

797 

1458 

1513 

1567 

1622 

1676 

1736 

1786 

1840 

I8y4 

1948 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

OF  NUMBERS 

17 

N. 

0 

903090 

1 

2 

3 

4 

6 

6 

7 

8 

9 

800 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

803 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

6148 

5202 

804 

5358 

5310 

6364 

5418 

5472 

5526 

5580 

6634 

5688 

5742 

805 

5796 

5860 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

806 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

808 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

809 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

810 

8485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

811 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

812 

9656 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

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813 

910091 

0144 

0197 

0251 

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0464 

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0944 

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1051 

1104 

815 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

816 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2115 

2169 

817 

2222 

2275 

2323 

2381 

2435 

2488 

2541 

2594 

2645 

2700 

818 

2753 

2808 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

819 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

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820 

3814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

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4398 

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4872 

4925 

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1010 

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1946 

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4021 

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4147 

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840 

4279 

4331 

4383 

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4693 

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4796 

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4951 

5003 

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5312 

5364 

5415 

5467 

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5725 

5776 

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5828 

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5931 

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6034 

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6137 

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6240 

6291 

844 

6342 

6394 

6445 

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6908 

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7011 

7062 

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7165 

7216 

7268 

7319 

846 

7370 

7422 

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7730 

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7832 

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7935 

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8037 

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8140 

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8293 

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848 

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8857 

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8959 

9010 

9081 

9112 

9163 

9216 

9266 

9317 

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J 

18 

I^OGARITHMS 

N. 
850 

0 

923419 

I  1  2 

3 

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5 

6 

7 

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9 

. 

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1000 

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1204 

1254 

1305 

1366 

1407 

864 

1458 

1509 

1660 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

856 

2474 

2624  2676 

2626 

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2727 

2778 

2829 

2879 

2930 

857 

2981 

3061 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

858 

•3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

869 

3993 

4044 

4094 

4145 

4195 

4246 

4269 

4347 

4397 

4448 

880 

4498* 

4549 

4699 

4650 

4700 

4751 

4801 

4852 

4902 

4950 

861 

5003 

5054 

5104 

6154 

6205 

6265 

5306 

6856 

6406 

6457 

862 

6507 

5558 

6608 

6658 

6709 

5759 

6809 

5860 

6910 

5960 

863 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

864 

6514 

6564 

6614 

6665 

6715 

6766 

6815 

6865 

6916 

6966 

866 

7016 

7066 

7117 

71-67 

7217 

7267 

7317 

7367 

7418 

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866 

7518 

7568 

76lg 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

867 

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8069 

8119 

8169 

8:^19 

8269 

8320 

8370 

8420 

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868 

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869 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

870 

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9616 

9669 

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9769 

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9869 

9918 

9968 

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0068 

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0168 

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0267 

0317 

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0417 

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0516 

0566  i  0616 

0866 

0/16 

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1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

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1462 

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1511 

1561 

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1859 

1909 

1968 

875 

2008 

2068 

2107 

2157 

2207 

2256 

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2405 

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2504 

2554 

2603 

2653 

2702 

2762 

2801 

2851 

2901 

2950  i 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445  1 

878 

3495 

3644 

3693 

3643 

3692 

3742 

3791 

3841 

3890 

3939  i 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4286 

4336 

4384 

4433  i 

1 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

881 

4976 

6026 

5074 

5124 

5173 

6222 

5272 

6321 

5370 

5419 

882 

5469 

6518 

5667 

6616 

5666 

5715 

6764 

6813 

6862 

6912 

883 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6364 

6403 

884 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

885 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

886 

7434 

7488 

7532 

7681 

7630 

7679 

7728 

7777 

7826 

7876 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8365 

888 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8765 

8804 

8863 

889 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

899 

9390 

9439 

9488 

9636 

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9731 

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9829 

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9878 

9926 

9975 

..24 

..73 

.121 

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.219 

.267 

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892 

950865 

0414 

0462 

0611 

0560 

0608 

0667 

0706 

0754 

0803 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

894 

1338 

1386 

1436 

1483 

1632 

1580 

1629 

1677 

1726 

1776 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

896 

2308 

2366 

2405 

2463 

2502 

2550 

2699 

2647 

6696 

2744 

897 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

898 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3616 

3663 

3711  i 

899 

3760 

3808 

8856 

3905 

3963 

.  4001 

4019 

40JS 

4146 

4194 

OF  NUMBERS.             ]9 

N. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

'  9 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

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4677 

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4726 

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4918 

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6014 

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5158 

902 

6207 

5256 

5303 

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6399 

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5592 

5640 

903 

6688 

5736 

5784, 

5832 

5880 

5928 

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6024 

6072 

6120 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6565 

6553 

6601 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

,  908 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8088 

908 

8086 

8134 

8181 

8229 

8277 

8326 

8373 

8421 

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8516 

909 

8564 

8612^ 

8659 

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8803 

8860 

8898 

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910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

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911 

9518 

9566 

9814 

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9804 

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9900 

9947  . 

. 

912 

9995 

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.138 

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913 

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0518 

0566 

0613 

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1041 

1089 

1136 

1184 

1231  • 

1279 

1326 

1374 

915 

1421 

1469 

1616 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

917 

2369 

2417 

2464 

2511 

2559 

2608 

2653 

2701 

2748 

2795 

918 

2848 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

919 

3316 

3363 

3410 

3457 

3604 

3552 

3699 

3646 

3693 

3741 

:  920 

3788 

3835 

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3929 

3977 

4024 

4071 

4118 

4165 

4212  ■ 

921 

4260 

4307 

4354 

4401 

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4495 

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4590 

4637 

4684 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

923 

5202 

5249 

5296 

5343 

6390 

5437 

5484 

5531 

5578 

5625 

924 

5672 

5719 

5766 

6813 

5860 

5907 

5954 

6001 

6048 

6095 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6617 

6564 

926 

6611 

6658 

6705 

6762 

6799 

6845 

6892 

6939 

6986 

7033 

927 

7080 

7127 

7173 

7220 

7267 

7314 

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7501 

928 

7548 

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7688 

7735 

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7829 

7875 

7922 

7969 

929 

8016 

8632 

8109 

8156 

8263 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

931 

8950 

8996 

9043 

9090 

.9136 

9183 

9229 

9276 

9323 

9369 

932 

9416 

9463 

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9556 

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9742 

9789 

9835 

933 

9882 

9928 

997^ 

..21 

..68 

.114 

.161 

.207 

.254 

.300 

934 

970347 

0393 

0440 

0486 

6533 

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6626 

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0765 

935 

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0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1564 

1601 

1647 

1693 

937 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

938 

2203 

2249 

229S 

2342 

2388 

2434 

2481 

2627 

2573 

2619 

939 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

8543 

941 

8590 

3636 

3682 

3728 

3774 

3820 

8866 

3913 

3959 

4005 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

943 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

944 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

945 

5432 

5478 

5524 

,5570 

5616 

5662 

5707 

5753 

5799 

5845 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

947 

6350 

6396 

6442 

6488 

6533 

6679 

6925 

6671 

6717 

6763 

948 

6803 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

949 

7266 

7312 

7358  7403 

7449 

7496 

7541 

7586 

7632 

7678 

' 

2a 

1 
LOGARITHMS 

N. 

0 

1 

S 

3 

4 

5 

6 

7 

8 

9 

950 

977724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

854G 

8591 

96-2 

8637 

8683 

8728 

8774 

8819 

8866 

8911 

8956 

9002 

9047 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

964 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9968 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

956 

0458 

0503 

0549 

0594 

0640 

0686 

0730 

0776 

0821 

0867 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

958 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

9G0 

2271 

2316 

2362 

2407 

2462 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3(H0 

3085 

3130 

962 

3176 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

963 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

964 

4077 

4122' 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932  I 

966 

4977 

5GG2 

5067 

6112 

6157 

5202 

5247 

6292 

6337 

5382 

967 

B426 

5471 

5516 

6561 

5606 

5651 

6699 

5741 

6786 

5830 

968 

5876 

5920 

6965 

6010 

6056 

6100 

6144 

6189 

6234 

6279 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6773 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

971 

7219 

7264 

7300 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

973 

8U3 

8157 

820Q 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

975 

90a5 

9049 

9093 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

&76 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.206 

.250 

.294  1 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738  1 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182  I 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625  j 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

3067  1 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509  1 

983 

2554 

2698 

2642 

:  2686 

2730 

2774 

2819 

2863 

2907 

2951  i 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392  1 

j 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833  j 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273  i 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

988 

4757 

4801 

4845 

4886 

4933 

,4977 

5021 

6065 

5108 

5152 

989 

6196 

6240 

6284 

5328 

5372 

;6416 

5460 

6504 

6547 

6691 

990 

6635 

6679 

5723 

5767 

6811 

5854 

6898 

5942 

6986 

6030 

991 

6074 

6117 

6161 

6205 

6249 

;6293 

6337 

6380 

6424 

6468 

992 

6612 

6655 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

6949 

6993 

7037 

7080 

7124 

:7168 

7212 

7255 

7299 

7343 

994 

7386 

7430 

7474 

7517 

7661 

7605 

7648 

7692 

7736 

7779  1 

1 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

996 

8259 

8303 

8347 

8390 

8434 

!8477 

8521 

8564 

8608 

8652 

997 

8695 

8739 

8792 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9622 

999 

9665 

9609 

9662 

9696 

9739 

9,783 

9826 

9870 

9913 

i 

9957 

♦if 


TABLE  II.    Log.  Sines 

Hud  Tangents,  (0°)  Natural  Sines.          *1 

T" 

Sine. 

D.IO" 

(Jos'iiie. 

in(F 

T«ng.  ■ 

D.IO" 

Coiang. 
Infinite. 

N.8ine. 

N.  COS. 

0 

0.000000 

10.000000 

0.000000 

00000 

lOOOOO 

60 

1 

6.463726 

000000 

6.463726 

13.536274 

00029 

100000 

59 

2 

764756 

000000 

764756 

235244 

00058 

lOOOOU 

58 

3 

940847 

000000 

940847 

059153 

00037 

100000 

57 

4 

7.085786 

000000 

7.065786 

12.934214 

00116 

100000 

56 

5 

162696 

000000 

162696 

837804 

00145 

100000 

56 

6 

241877 

9.999999 

241878 

758122 

00175 

100000 

54 

7 

308824 

999999 

308825 

691175 

00204 

100000 

63 

8 

366816 

999999 

366817 

633183 

00233 

I 00000 

62 

9 

417968 

999999 

417970 

582030 

00262 

100000 

51 

10 

463725 

999998 

463727 

536273 

00291 

100000 

50 

11 

7.505118 

9.999998 

7.505120 

12.494880 

00320 

99999 

49 

12 

54290S 

999997 

542909 

457091 

00349 

99999 

48 

13 

677668 

999997 

577672 

422328 

00378 

99999 

47 

14 

609853 

999996 

609867 

390143 

00407 

99999 

46 

15 

639816 

999996 

639820 

360180 

00436 

99999 

45 

16 

667845 

999996 

667849 

332151 

00465 

99999 

44 

17 

694173 

999995 

694179 

305821 

00496 

99999 

43 

18 

718997 

999994 

719003 

280997 

00524 

99999 

42 

19 

742477 

999993 

742484 

257516 

00553 

99998 

41 

20 

764754 

999993 

764761 

236239 

00582 

99998 

40 

21 

7.785943" 

9.999992 

7.785951 

12.214049 

00611 

99998 

39 

22 

806146 

999991 

806155 

193845 

00640 

99998 

38 

23 

825451 

999990 

825460 

174540 

00669 

99998 

37 

24 

843934 

999989 

843944 

156056 

00698 

99998 

36 

25 

861663 

999988 

861674 

138326 

00727 

99997 

35 

26 

878695 

999988 

878708 

121292 

00756 

99997 

34 

27 

895085 

999987 

896099 

104901 

00785 

99997 

33 

28 

910879 

999986 

910894 

039106 

00814 

99997 

32 

29 

926119 

999985 

926134 

073866 

00844 

99996 

31 

30 

940842 

999983 

940858 

059142 

00873 

99996 

30 

31 

7.955082 

9.999982 

7.956100 

2298 
2227 
2161 
2098 
2039 
1983 
1930 
1880 
1833 
1787 
1744 
1703 
1664 
1627 
1591 
1557 
1524 
1493 
1463 
1434 
1406 
1379 
1353 
1328 
1304 
1281 
1259 
1238 
1217 

12.044900 

00902 

99996 

29 

32 

968870 

2298 

S99981 

0.2 

968889 

031111 

00931 

99996 

28 

33 

982233 

2227 

999980 

0.2 

982253 

017747 

00900 

99995 

27 

34 

995198 

2161 

999979 

0.2 

996219 

004781 

00989 

99995 

26 

35 

8.007787 

2098 

999977 

0-2 

8.007809 

11.992191 

01018 

99995 

25 

36 

020021 

2039 

999976 

0-2 
0-2 

020045 

979955 

01047 

99995 

24 

37 

031919 

1983 

999975 

031945 

968055 

01076 

99994 

23 

38 

043501 

1930 
1880 
1832 
1787 
1744 
1703 
1664 
1626 
1691 
1567 
1524 
1492 
1462 
1433 
1405 
1379 
1353 
1328 
1304 
1281 
1269 
1237 
1216 

999973 

0-2 
0-2 

043527 

956473 ' 

01105 

99994 

22 

39 

054781 

999972 

054809 

945191  1 
934194 1 

01134 

99994 

21 

40 

065776 

999971 

0*2 

065808 

01164 

99993 

20 

41 

8.076500 

9.999969 

0'2 
0-2 

8.076531 

11.923469 

01193 

99993 

19 

42 

086965 

999968 

086997 

913003  i 

01222 

99993 

18 

43 

097183 

999966 

0'2 

097217 

902783 1 

01251 

99992 

17 

44 

107167 

999964 

0'2 

o;3 

107202 

892797 1 

01280 

99992 

16 

45 

116926 

999963 

116963 

883037 

01309 

99991 

15 

46 

126471 

999961 

0  3 
0.3 
0.3 

126610 

873490 

01338 

99991 

14 

47 

136810 

999959 

135851 

864149 

01367 

1)9991 

13 

48 

144953 

999958 

144996 

855004 

01396 

99990 

12 

49 

153907 

999956 

0.3 
0.3 

163952 

846048 

01425 

99990 

11 

50 

162681 

999954 

162727 

837273 

01454 

99989 

10 

51 

8.171280 

9,999952 

0.3 
0.3 
0.3 
0.3 
0.3 
0.3 

8.171328 

11.828672 

01483 

99989 

9 

52 

179713 

999950 

179763 

820237 

01513 

99989 

8 

53 

187985 

999948 

188036 

811964 

01542 

99988 

7 

54 

196102 

999946 

196166 

803844 

01571 

99988 

6 

56 

204070 

999944 

204126 

795874' 

01600 

99987 

5 

56 

211895 

999942 

211953 

7&8047 

01629 

99987 

4 

57 

219581 

999940 

0.4 
0.4 

219641 

780359  j 

01658 

99986 

3 

58 

227134 

999938 

227195 

772805 ' 

01687 

9998() 

2 

59 

234557 

999936 

0.4 
0.4 

234621 

765379 

01716 

99985 

1 

60 

241855 

999934 

241921 

758079:,  01746 

99985 

Cosine. 

Sine. 

Coihmg:. 

Tnnar.  '  N.  cos. 

N.  sine- 

89  Degrees.                            | 

22 


hog.  Sines  aud  TangentB.  (l")  Natural  Sines.   TABLE  H. 


D.IO' 


Cosine. 


D.IO' 


Tang. 


DIG"  Coiang. 


N.  sine.  N.  cos. 


10 

11 

12 

13 

14 

16 

16 

17 

18 

19 

20 

21 

22 

23 

24 

26 

26 

27 

28 

29 

30 

31  8 

32 

33 

34 

36 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

60 

61 

62 

53 

54 

66 

56 

57 

68- 

69 

60 


.241856  jj  g 

249033  „,, 

256094  I;'' 

26304-2  };°° 

2€9881  {^^ 

276(il4  {{r.'t 

283243  |iX« 

289773  V^l 

290207  \ali 

302546  \lf, 

308794  \Yl^ 

.314954  }^, ' 

321027  aoi 

327016  QOR 

332924  g^j 

338763  q-q 

344604  all 

350181  qo^ 

355783  Qoo 

361316  Qin 

366777  gg" 

.372171  S^o 

377499  S°? 

382762  ^^ 

387962  ^L 

393101  o2^ 

398179  o^ 

403199  007 

408161  ofo 

413068  or^ 

417919  oXX 

.422717  ?^J 

427462  7^2 

432156  7^f 

436800  7^R 

441394  7^0 

445941  III 

450440  740 

454893  7^ 

459301  ^tj 

463665  «on 

.467985  ':" 

472263  ;l-\f 

476498  gXX 

480693  ^^l 

484848  ^«fi 

488963  l^l 

493040  ^', 
497078!  ^^5 
501080  i  l^i 

506045  i  °^^ 

.608974  I  °5q 

612867  !  ^11 

516726  ^^^ 

620551  ;  lii 

524343  2^^ 

528102  ^^^ 
531828 :  fi,^ 

535623  ^}° 

539186  «i.^ 
542819  .  ^"^ 


9.999934 
999932 
999929 
999927 
999925 
999922 
999920 
999918 
999916 
999913 
999910 

9.999907 
999905 
999902 


999897 
999894 
999891 
999888 


9.999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
999854 
999851 

9.999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 

9.999812 
999809 
999805 
999801 
999797 
999793 
999790 
999786 
999782 
999778 

9.999774 
999769 
999765 
999761 
999757 
999753 
999748 
999744 
999740 
999735 


0.4 
0.4 
0.4 
0.4 
0.4 
0.4 
0.4 


0.5 
0.5 
0.6 
0.5 
0.6 
0.5 
0.5 
0.5 
0-6 
0-6 
0.6 
0.6 
0-6 
0.6 
0.6 
0.6 
0.6 
0.6 
0.6 
0-6 
0-6 
0.6 
0.6 
0-7 
0-7 
0.7 
0.7 
0.7 
0.7 


.241921 
249102 
266166 
263115 
269956 
276691 
283323 
289866 
296292 
302634 


.3160-46 
321122 
327114 
333026 


344610 
350289 
356895 
361430 


.372292 
377622 


8ine. 


388092 
393234 
398316 
403338 
408304 
413213 
418068 

.422869 
427618 
432315 
436962 
441560 
446110 
450613 
455070 
469481 
463849 

.468172 
472454 
476693 
480892 
485050 
489170 
493250 
497298 
501298 
505267 

.509200 
513098 
516961 
620790 
624586 
528349 
532080 
536779 
539447 
643084 

Cotarifr- 


1197 
1177 
1168 
1140 
1122 
1105 
1089 
1073 
1057 
1042 
1027 
1013 
999 
985 
972 
959 
946 
934 
922 
911 


879 
867 
857 
847 
837 
828 
818 
809 
800 
791 
783 
774 
766 
758 
750 
743 
736 
728 
720 
713 
707 
700 
693 
686 
680 
674 
668 
661 
655 
650 
644 
638 
633 
627 
622 
616 
611 
606 


11.758079 


743835 
736885 
730044 
723309 
716677 
710144 
703708 
697366 
691116 

11.684954 
678878 
672886 
666975 
661144 
665390 
649711 
644106 
638670 
633105 

11-627708 
622378 
617111 
611908 
606766 
601685 
596662 
591696 
686787 
581932 

11.577131 


567685 
563038 
568440 
553890 
549387 
544930 
540519 
636151 

11.531828 
527546 
523307 
519108 
614950 
610830 
506750 
502707 
498702 
494733 

11.490800 
486902 
483039 
479210 
476414 
471651 
467920 
464221 
460563 
456916 


01742 
01774 
01803 
01832 
01862 
01891 
01920 
01949 
01978 
02007 
02036 
02065 
02094 
02123 
02152 
02181 
02211 
02240 
02269 
02298 
02327 
02356 
02385 
02414 
02443 
02472 
02501 
02630 
02660 
02589 
02618 
02647 
02676 
02705 
02734 
02763 
02792 
02821 
02850 


99985 
99984 
99984 
99983 
99983 
99982 
99982 
99981 


99980  62 


99980 
99979 
99979 
99978 
99977 
99977 
99976 
99976 
99975 
99974 
99974 
99973 
99972 
99972 
99971 
99970 
99969 
99969 
99968 
99967 
99966 


99.9661  30 

99965 

99964 

99963 

99963 

99962 

99961 

99960 

99959 


02879  99959 


99958 
99957 
99956 
99955 
99954 
99953 
99952 
99952 
99951 
99950 
99949 
03228  99948 
03257  99947 
03286  99946 
03316  99945 
03346  99944 
03374  99943 
03403  99942 


02908 

02938 

02967 

02996 

03025 

03054 

03083 

03112  99952 

03141 

03170 

03199 


03432 
03461 
03490 


99941 
99940 
99939 


Tang. 


N.  COS.  N.8ine 


88  Degrees. 


TABLE  II.        Log.  Sines  and  Tangents.    (2°)     Natural  Sines. 


23 


8.542819 
54G4'22 
549995 
553539 
557054 
560540 
563999 
567431 
570836 
574214 
577666 

8.580892 
584193 
587469 
590721 
593948 
597152 
600332 
603489 
606623 
609734 

8.612823 
615891 
618937 
621962 
624965 
627948 
630911 
633854 
636776 
639680 

8.642563 
645428 
648274 
651102 
653911 
656702 
659475 
662230 
664968 
667689 

8.670393 
673080 
675751 
678405 
681043 
683665 
686272 


691438 
693998 
8.696543 
699073 
701589 
704090 
706577 
709049 
711507 
713952 
716383 
_718800 
Cosme. 


D.  10" 


600 
595 
691 
586 
581 
576 
572 
567 
563 
559 
554 
550 
546 
542 
538 
634 
530 
526 
522 
519 
515 
511 
508 
504 
501 
497 
494 
490 
487 
484 
481 
477 
474 
471 
468 
465 
462 
469 
456 
453 
451 
448 
445 
442 
440 
437 
434 
432 
429 
427 
424 
422 
419 
417 
414 
412 
410 
407 
405 
403 


Cosine. 

1.999735 
999731 
999726 
999722 
999717 
999713 
999708 
999704 


999694 


999685 
999680 
999675 
999670 
999665 
999660 
999655 
999550 
999645 
999640 
999635 
999629 
999324 
999619 
999614 
999608 
999603 
999597 
999592 
999586 
999581 
999575 
999570 
999564 
999558 
999553 
999547 
999541 
999535 
999529 
,999524 
999518 
999512 
999506 
999500 
999493 
999487 
999481 
999475 
999469 
,999463 
999456 
999450 
999443 
999437 
999431 
999424 
999418 
999411 
999404 


D.  10" 


Tang. 


0.7 
0.7 
0.7 
0-8 
0-8 
0-8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.8 
0.9 
0.9 
0.9 
0.9 
0.9 
0.9 
0-9 
0.9 
0-9 
0.9 
0-9 
0.9 
0.9 
0.9 
0.9 
1.0 
1-0 
1.0 
1.0 
1-0 
1.0 
1.0 
1.0 
1-0 
1-0 
1.0 
1.0 
1.0 
1.0 
1.0 
1.0 


1.543084 
546691 
550268 
553817 
557335 
560828 
564291 
567727 
571137 
574520 
577877 

.581208 
584514 
587795 
591051 
594283 
597492 
600677 
603839 
606978 
610094 

.613189 
616262 
619313 
622343 
625352 
628340 
631308 
634256 
637184 
640093 

,642982 
645853 
648704 
651537 
664352 
657149 
659928 
662689 
665433 
668160 

.670870 
673563 
676239 
678900 
681544 
684172 
6-6784 
689381 
691963 
694529 

.697081 
699617 
702139 
704246 
707140 
709618 
702083 
714534 
716972 
719396 

Coians:. 


D.  10"i  Coiang.  |{N.  sine.  N.  cos. 


602 
593 
591 
587 
582 
577 
573 
568 
564 
559 
555 
551 
547 
543 
539 
535 
631 
527 
523 
519 
516 
512 
508 
505 
501 
498 
495 
491 
488 
485 
482 
478 
475 
472 
469 
466 
463 
460 
457 
454 
453 
449 
446 
443 
442 
438 
485 
433 
430 
428 
425 
423 
420 
418 
415 
413 
411 
408 
406 
404 


11.456916 
453309 
449732 
446183 
442664 
439172 
435709 
432273 
428863 
425480 
422123 

11.418792 
415486 
412205 
408949 
405717 
402508 
399323 
396161 
393022 
389906 

11.386811 
383738 
380687 
377657 
374648 
371660 
368692 
365744 
362816 
359907 

11.357018 
354147 
351296 
348463 
345648 
342851 
340072 
337311 
334567 
331840 

11.329130 
326437 
323761 
321100 
318456 
315828 
313216 
310619 
308037 
305471 

11.302919 
300383 
297861 
296354 
292860 
290382 
287917 
285465 
283028 
280604 


03490 


03519  99938 


03548 


03577  99936 


03606 
1 03635 
1 03664 
■03693 
103723 
03752 
1 03781 
03810 
03839 
03868 
03897 
03926 
03955 
03984 
04013 
04042 
104071 
104100 
103129 
{04159 
04188 
04217 
04246 
04275 
04304 
04333 
04362 
04391 
04420 
04449 
04478 
04507 
04536 
04565 
04594 
04623 
04653  99892 
04682  99890 
0471199889 
04740  99888 
04769  99886 
04798  99885 


Tans 


04827 


04856  9988 


04885 
04914 
04943 
04972 
05001 
05030 
05059 
05088 
05117 
05146 
05176 
06205 
05234 


99939 


99937 


99935 
99934 
99933 
99932 
99931 
99930 
99929 
99927 
99926 
99926 
99924 
99923 
99922 
99921 
99919 
99918 
99917 
99916 
99915 
99913 
99912 
99911 
99910 
99909 
99907 
99906 
99906 
99904 
99902 
99901 
99900 


99897 
99896 
99894 


99883 


99881 
99879 
99878 
99876 
99876 
99873 
99872 
99870 
99869 
9y867 


99864 
99863 


N.  COS.  N.8ine 


87  Degrees. 


24 


iOg.  Sines  and  Tangciiis.     (3°;    Natural  Sines,        TABLE  II. 


0 

2 

3 

4 

5 

6 

7 

8 

9 
10 
11 
•12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
28 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
51  8 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Cosine. 


1 

1 

1. 

1 

1 

1 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 


Sine.  |U.  W     Cosine.  D71o 

.71880a 
721204 
723595 
725972 
728337 
730{>88 
733027 
735354 
737667 
739969 
742259 
.744536 
746802 
749055 
751297 
753528 
755747 
757955 
760151 
762337 
764511 
.766675 
768828 
770970 
773101 
776223 
777333 
779434 
781524 
783605 
785675 
.787736 
789787 
791828 
793859 
795881 
797894 
799897 
801892 
803876 
805852 
,807819 
809777 
811726 
813667 
815599 
817522 
819436 
821343 
823240 
825130 
,827011 
828884 
830749 
832607 
834456 
836297 
838130 
839956 
841774 
843586 


401 
398 
396 
394 
392 
390 
388 
386 
384 
382 
380 
378 
376 
374 
372 
370 
368 
366 
364 
362 
361 
359 
357 
355 
353 
352 
350 
348 
347 
345 
343 
342 
340 
339 
337 
336 
334 
332 
331 
329 


325 
323 
322 
320 
319 
318 
316 
316 
313 
312 
311 
309 
308 
307 
306 
304 
303 
302 


.999404 
999398 
999391 
999384 
999378 
999371 
999364 
999357 
999350 
999343 
999336 
.999329 
999322 
999315 
999308 
999301 
999294 
999286 
999279 
999272 
999265 
.999257 
999250 
999242 
999235 
999227 
999220 
999212 
999205 
999197 
999189 
.999181 
999174 
999166 
999158 
999150 
999142 
999134 
999126 
999118 
999110 
.999102 
999094 
999086 
999077 


999061 
999053 
999044 
999036 
999027 
.999019 
999010 
999002 
998993 
998984 
998976 
998967 


998950 
998941 


1.2 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
13 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 
1.3 


1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.5 
1.5 
1.5 


lanj; 


.719396 
721806 
724204 
726588 
728959 
731317 
733663 
735996 
738317 
740326 
742922 
,745207 
747479 
749740 
751989 
754227 
766453 
758668 
760872 
763065 
765246 
767417 
769578 
771727 
773866 
775995 
778114 
780222 
782320 
784408 
786486 

8.788554 
790613 
79-2662 
794701 
793731 
798752 
80'J763 
802765 
804858 
80  )742 
808717 
810683 
81-2641 
814589 
816529 
818461 
820384 
822298 
824205 
826103 

8.827992 
829874 
831748 
833613 
835471 
837321 
839163 
840998 
842826 
844644 


Cotanp. 


l».  !(/ 


402 
399 
397 
395 
393 
391 
389 
387 
385 
383 
381 
379 
377 
375 
373 
371 
3o9 
367 
365 
364 
362 
360 
358 
356 
355 
353 
351 
350 
348 
346 
345 
343 
341 
340 
338 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
316 
314 
312 
311 
310 
308 
307 
306 
304 
303 


Cotang.  |(N.  sine. 


11.2806041 
278194] 
276796 
273412 
271041  I 
268683  i 
266337  I 
264004  I 
261683  I 
259374;  1 05496 
257078  I  05524 


05234 
05263 
05292 
06321 
05350 
05379 
05408 
06437 
05466 


11.254793 
252521 
250260 
248011 
245773 
243547 
241332 
239128 
236935 
234754 

11.232583 
230422 
228273 
226134 
224005 
221886 
219778 
217680 
215592 
213514 

11.211446 
209387 
207338 
205299 
203269 
201248 
199237 
197235 
195242 
193258 

11.191283 
189317 
187359 
185411 
183471 
181639 
179616 
177702 
175796 
173897 

11.172008 
170126 
168252 
166387 
164529 
162679 
160837 
159002 
167175 
156366 


05553 

05582 

05611 

05640 

05669 

05698 

0572 

05766 

05785 

05814 

05844 

05873 

05902 

05931 

05960 


Tang. 


06018 
06047 
06076 
06105 
06134 
06163 
06192 
06221 
06250 
06279 
06308 
06337 
06366 
06395 
08424 
06463 
06482 
06511 
06540 
06569 
06598 
06627 
06656 
06685 
08714 
06743 
06773 
06802 
06831 
06860 
06889 
06918 
06947 
06976 
'iIn.  cos 


N.cos. 


99863 

99861 

99860 

99858 

99857 

99855 

99854 

99852 

99851 

99849 

99847 

99846 

99844 

99842 

99841 

99839 

99838 

99836 

99834 

99833 

99831 

99829 

9982' 

99826 

99824 

99822 

99821 

99819 

99817 

99815 

99813 

99812 

99810 

99808 

99806 

99804 

99803 

99801 

99799 

99797 

99795 

99793 

99792 

99790 

99788 

99786 

99784 

99782 

99780 

99778 

99776 

99774 

99772 

99770 

99768 

99766 

99764 

99762 

99760 

99758 

99766 


.V.Bine 


86  Degrees. 


TABLE  11. 


.og.  Sines  and  Tangents.    (4°)    Natural  Sines. 


25 


D.  W 

300 

299 
298 
297 
295 
294 
293 
292 
291 
2;)0 
288 
287 
286 
285 
284 
283 
282 
281 
279 
279 
277 
276 
275 
274 
273 
272 
271 
270 
269 
268 
267 
266 
265 
264 
263 
262 
261 
260 
259 
258 
257 
257 
256 
255 
254 
253 
252 
251 
250 
249 
249 
248 
247 
246 
245 
2'14 
243 
243 
242 
241 


Cosine. 

9.998941 
998932 
998923 
998914 
998905 
998896 
998887 
998878 
998869 
998860 
998851 

9.998841 
998832 
998823 
998813 
998804 
998795 
998785 
998776 
998766 
998757 

9.998747 
998738 
998728 
998718 
998708 


8.843586 
845387 
847183 
848971 
850751 
852525 
854291 
856049 
857801 
859546 
861283 

8.863014 
864738 
866455 
868165 
869868 
871565 
873255 
874938 
876615 
878285 

8.879949 
881607 
883258 
884S03 
886542 
888174 
889801 
891421 
893035 
894643 
.896246 
897842 
899432 
901017 
902596 
904169 
905736 
907297 
908853 
910404 

8.911949 
913488 
915022 
916550 
918073 
919591 
921103 
922610 
924112 
926609 

8.927100 
928587 
930068' 
931544 
933015 
934481 
935942 
937398 
938850 
940296 
Cosine. 


998689 
998679 
998669 
998659 
9.998649 
998639 
998629 
998619 
998609 


998589 
998578 
998568 
998558 
9.998648 
998537 
998627 
998616 
998506 
998495 
998485 
998474 
998464 
998453 
998442 
998431 
998421 
998410 
998399 


998377 
998366 
998355 
998344 


Sine. 


D.  10" 


Tang, 


D.  10"      Cotang.     l|N.  sine.  N.  cos 


884530 
886185 
887833 
889476 
891112 
892742 
894366 
895984 

.897596 
899203 
900803 
902398 
903987 
906570 
907147 
908719 
910285 
911846 

.913401 
914951 
916495 
918034 
919668 
921096 
922619 
924136 
925649 
927156 

.928658 
930155 
931647 
933134 
934616 
936093 
937565 


940494 
941952 
Cotang. 


302 
801 
299 
298 
297 
293 
295 
293 
292 
291 
290 
289 
288 
287 
285 
284 
283 
282 
281 
280 
279 
278 
277 
276 
275 
274 
278 
272 
271 
270 
269 
268 
267 
266 
265 
264 
263 
262 
261 
260 
259 
258 
267 
256 
256 
255 
264 
263 
252 
251 
250 
249 
249 
248 
247 
246 
245 
244 
244 
243 


11 


11.166366 
153646 
151740 
149943 
148164 
146372 
144697 
142829 
141068 
139314 
137667 
,  135827 
134094 
132368 
130649 
128936 
127230 
125531 
123838 
122151 
120471 
118798 
117131 
116470 
113815 
112167 
110524 
108888 
107258 
106634 
10-1016 
102404 
100797 
099197 
097602 
096013 
094430 
092863 
091281 
089715 
088154 

11.086599 
085049 
083606 
081966 
080432 
078904 
077381 
075864 
074361 
072844 

11.071342 


06976 
07005 
07034 
07063 
07092 
07121 
07150 
07179 
07208 
07237 
07266 
07295 
07324 
07363 
07382 
07411 
07440 
07469 
07498 
07527 
07556 
07586 


99756 
99754 
99752 
99750 
99748 
99746 
99744 
99742 
99740 
99738 
99736 
99734 
99731 
99729 
99727 
99725 
99723 
99721 
99719 
99716 
99714 
99712 


07614  99710 


1 07643 
07672 
07701 
07730 
07759 
07788 
07817 
07846 
07875 
07904 
07933 
07962 
07991 
08020 
08049 
08078 
08107 
08136 
08165 


08223 
08252 
08281 
08310 


108368 
08397 
08426 
08455 
1 08484 
108513 
1 08542 
i 08671 
108600 


068863 

066866 

065384 

063907 

06243511 08629  |9y62 

060968 

059606 

068048 


Tang. 


08658 
08687 


08716  99619 


N.  co,«.  -N.eine 


99708 
99705 
99703 
99701 
99699 
99696 
99694 
99692 
99689 
99687 
99685 
99683 
99680 
99678 
99676 
99673 
d9671 
99668 
99666 


08194  99664 


99661 
99659 
99657 
99654 
99652 
99649 
99647 
99644 
99642 
99639 
99637 
99635 
99632 
99630 


99625 
99622 


85  Degrees. 


26 


hog.  Sines  and  Tangents.  (5°)  Natural  Sines.   TABLE  II. 


0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

56 

5] 

62 

63 

54 

65 

56 

57 

58 

69 

60 


Sine. 

8.940296 
941738 
943174 
944608 
946034 
947456 
948874 
950287 
95169S 
953100 
954499 
955894 
957284 
958670 
960052 
901429 
962801 
984170 
965534 
966893 
968249 

8.969600 
970947 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 

8.982883 
984189 
985491 
986789 


D.  10" 


989374 
990660 
991943 
993222 
994497 

B. 995768 
997036 
998299 
999560 

9.000816 
002069 
003318 
004563 
005805 
007044 

9.008278 
009510 
010737 
011962 
013182 
014400 
015613 
016824 
018031 
019235 
Cosine. 


240 

239 

239 

238 

237 

236 

235 

235 

234 

233 

232 

232 

231 

230 

229 

229 

228 

227 

227 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

214 

214 

213 

212 

212 

211 

211 

210 

209 

209 

208 

208 

207 

206 

206 

203 

205 

204 

203 

203 

202 

202 

201 

201 


Cosine. 


(9.998344 
998333 
998322 
998311 
998300 
998289 
998277 
998266 
998255 
998243 
998232 
9.998220 
998209 
998197 
998186 
998174 
998163 
998151 
998139 
998128 
998116 
9.998104 
998092 
998080 
998068 
998056 
998044 
998032 
998020 
998008 
997996 
9.997984 
997972 
997959 
997947 
997935 
997922 
997910 
997897 
997885 
997872 
,997860 
997847 
997835 
997822 
997809 
997797 
997784 
997771 
997758 
997745 
997732 
997719 
997706 
997693 
997680 
997667 
997654 
997641 
997628 
997614 


D.  10"j  Tang. 


1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

2.0 

2.0 

2,0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2,1 

2.1 

2,1 

2.1 

2,1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2,1 

2.1 

2.1 

2.1 

2.1 

2.2 

2.2 

2.2 

2.2 

2.2 

2.2 


Sine. 


941952 
943404 
944852 
946295 
947734 
949168 
950597 
952021 
953441 
954856 
956267 
8.957674 
959075 
960473 
961866 
963255 
964639 
906019 
967394 
968766 
970133 
.971496 
972855 
974209 
975560 
976906 
978248 
979586, 
980921 
982251 
983577 
8.984899 
986217 
987532 
988842 
990149 
991451 
992750 
994045 
995337 
996624 
.997908 
999188 
,000465 
001738 
003007 
004272 
005534 
006792 
008047 
009298 
010546 
011790 
013031 
014268 
015502 
016732 
017959 
019183 
020403 
021620 


D.  10"!  Cotang. 


Co  tang. 


242 

241 

240 

240 

239 

238 

237 

237 

236 

235 

234 

234 

233 

232 

231 

231 

230 

229 

229 

228 

227 

226 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

210 

216 

215 

215 

214 

213 

213 

212 

211 

211 

210 

210 

209 

208 

208 

207 

207 

206 

206 

205 

204 

204 

203 

203 


11.058048 
056596 
055148 
053705  i 
052266  I 
050832  I 
049403  i 
047979 
046559  I 
0451441 
043733  I 
11.042326! 
040925  i 
039527 
038134 
036745 
035861 
033981 
032606 
031234 
029867 
11.028504 
027145 
025791 
024440 
023094 
021752 
020414 
019079 
017749 
016423 
11.015101 
013783 
012468 
011158 
009851 
008549 
007250 
005955 
004663 
003376 
11.002092 
000812 
10.999535 
998262  I 
996993  I 
996728  I 
994466  I 
993208  I 


N.  sine 


08716 
08745 
08774 
08803 
08831 
08860 


08918 
08947 
08976 
09005 
09034 
09063 
09092 
09121 
09150 
09179 
09208 
09237 
09266 
09295 
09324 
09353 


99619  60 


99617 
99614 
99612 
99609 
99607 
99604 
99602 
99599 
99596 
99594 
99591 
99588 
99586 
99583 
99580 
99578 
99575 
99572 
99570 
99567 
99564 
99562 


09382|99559 
0941199556 
09440J99553 
09469199551 
09498199548 
09527199545 
09556199542 
09585i99540 
99537 
99534 
99531 
99528 
99526 
99523 
99520 
99517 


09614 
09642 
09671 
09700 
09729 
06758 
09787 
09816 

09845|99514 

09874199511 

09903199508 

09932  j9950i 

09961 J99503 

09990199500 

10019J9949 

10048J99494 

10077199491 

10106199488 

991953  1110135  99485 

990702  1 110164199482 

10.989454 


988210 

686969 

986732 

984498 

983268 

983041 

980817 

979597  j 

978380  I 


i  10192  99479 
11022199470 
110250,99473 
110279,99470 
10308  99467 


Tang. 


10337 
10366 
10395 
10424 
10463 


99464 
99461 
99458 
99455 
99452 
N.  COS.  Njsine. 


59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
^6 
26 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
6 
4 


8\   Degrees. 


TABLE  IT. 


Log.  Sines  and  Tangents.    (6"')    Natural  Sines. 


27 


60 


Sine. 

9.019235 
020436 
021632 
022825 
024016 
025203 
026386 
027567 
028744 
029918 
031089 

9.032257 
033421 
034582 
035741 
036896 
038048 
039197 
040342 
041485 
042625 
.043762 
044895 
046026 
047154 
048279 
049400 
050619 
051635 
052749 
053859 

9.054966 
056071 
057172 
058271 
059367 
060460 
061561 
062639 
063724 
064806 

9.066886 
066962 
068036 
069107 
070176 
071242 
072306 
073366 
074424 
075480 
076633 
077583 
078631 
079670 
080719 
081759 
082797 
083832 
084864 
086894 


I>.  10"[  Cosine.  ID.  10' 


Cosine. 


200 
199 
199 
198 
198 
197 
197 
196 
196 
195 
196 
194 
194 
193 
192 
192 
191 
191 
19a 
190 
189 
189 
180 
188 
187 
187 
186 
186 
185 
185 
184 
184 
184 
183 
183 
182 
182 
181 
J81 
180 
180 
179 
179 
179 
178 
178 
177 
177 
176 
176 
175 
176 
176 
174 
174 
173 
173 
172 
172 
172 


.997614 
997601 
997588 
997574 
997661 
997647 
997534 
997520 
997507 
997493 
997480 

.997466 
997452 
997439 
997425 
997411 
997397 
997388 
997369 
997355 
997341 

.997327 
997313 
997299 
997285 
997271 
997267 
997242 
997228 
997214 
997199 

.997185 
997170 
997156 
997141 
997127 
997112 
997098 
997083 
997068 
997053 

.997039 
997024 
997009 
996994 
996979 
996964 
996949 
996934 
996919 
996904 

.996889 
996874 
996858 
996843 
996828 
996812 
996797 
996782 
996766 
996761 


Sine 


I  2.2 
I  2.2 
2.2 
2.2 
2.2 
I  2.2 
2.3 
I  2.3 
!  2.3 
2-3 
2-3 
2.3 
2-3 
2.3 
2.3 
2.3 
2-3 
2-3 
2-3 
2-3 
2.3 
2.4 
2-4 
2.4 
2.4 
2-4 
2.4 
2.4 
2.4 
2-4 


2 

2 

2 

2 

2 

2 

2 

2 

2.5 

2.5 

25 

25 

2-5 

2.6 

2.6 

2-5 

2-6 

2.5 

2.6 

2.6 

2.5 

2.6 

2.6 

2.6 

2.5 

2.5 

2.6 

2  6 

2.6 

2.6 


Tang.  ,J>.  W' 


9.021620 
022834 
024044 
025251 
026455 
027655 
028852 
030046 
031237 
032425 
033609 

9,034791 
035969 
037144 
038316 
039485 
040651 
041813 
042973 
044130 
045284 

9.046434 
047682 
04872/ 
049869 
051008 
052144 
053277 
054407 
056635 
056659 

9.057781 
058900 
060016 
061130 
062240 
063348 
064453 
066656 
066655 
067762 

9.068846 
069038 
071027 
072113 
073197 
074278 
075356 
076432 
077505 
078576 

9.079644 
080710 
081773 
082833 
083891 
084947 
086000 
087050 


202 
202 
201 
201 
200 
199 
199 
198 
198 
197 
197 
196 
196 
195 
195 
194 
194 
193 
193 
192 
192 
191 
191 
190 
190 
189 
189 
188 
188 
187 
187 
186 
186 
185 
186 
185 
184 
184 
183 
183 
182 
182 
181 
181 
181 
180 
180 
179 
179 
178 
178 
178 
177 
177 
176 
176 
175 
175 
176 
174 


99354 
99361 
99347 
99344 
99341 
99337 
99334 
99331 
99327 
99324 
99320 
99317 
99314 
99310 
99307 
99303 
99300 
99297 
99293 
99290 
99286 
99283 
99279 
99276 
99272 
99269 
992G5 
99262 
99258 
y9256 
I   Tang.   Il  N.  cos.  N.sine. 


0453 
0482 
0511 
0540 
0569 
0597 
0626 
0655 
0684 
0713 
0742 
0771 
0800 
0829 
0858 
0887 
0916 
0945 
0973 
1002 
1031 
1060 
1089 
1118 
1147 
1176 
1205 
1234 
1263 
1291 


99452 
99449 
99446 
99443 
99440 
99437 
99434 
99431 
99428 
99424 
99421 
99418 
99415 
99412 
99409 
99406 
99402 
99399 
99396 
99393 
99390 


Cotang.  t  N.  sine.  N.  ccs. 

10.978380! 

977166  I 

975956  I 

974749  I 

973645  I 

972345  j 

971148 i 

969954 

968763  I 

967676  i 

966391  I 
10.965209 

964031 

962856 

961684 

960516 

959349 

958187 

957027 

955870 

954716 
10.963566 

952418 

951273 

950131 

948992 

947856 

946723 

945693 

944465 

943341 
10.942219 

941100 

939984 

938870 

937760 

936652 

935647 

934444 

933345  I 

932248 
10.9311541 

930062 

928973 

927887 

926803 

926722 

924644 

923668 

922496 

921424 
10.920356 

91929a 

»18227 

917167 

916109 

915053 

914000 

912950 

911902  jl 

910856 


99383 
99380 
99377 
99374 
99370 
99367 
99364 
99360 
1320'99357 


1349 
1378 
1407 
1436 
1465 
1494 
1523 
1552 
1580 
1609 
1638 
1667 
1696 
1726 
1754 
1783 
1812 
1840 
1869 
1898 
1927 
1956 
1985 
2014 
2043 
2071 
2100 
2129 
2168 
2187 


28 


Log.  Sines  and  Tangents.  (7°)  Natural  Sines.     TABLE  II. 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
61 
62 
53 
54 
55 
56 
57 
58 
59 
60 


D.  Ml    Cosine 


9.085894 
086922 
087947 
088970 
089990 
091008 
092024 
093037 
094047 
09505G 
096062 
9.097065 
098036 
099065 
100052 
101056 
102048 
103037 
104025 
105010 
105992 
9.106973 
107951 
108927 
109901 
110873 
111842 
112809 
113774 
114737 
115698 
.116656 
117613 
118567 
119519 
120469 
121417 
122362 
123306 
124248 
125187 
9.126126 
127060 
127993 
128925 
129864 
130781 
131706 
132630 
133551 
134470 
9.135387 
136303 
137216 
138128 
139037 
139944 
140860 
141754 
142655 
143565 


Cosine. 


171 

171 

170 

170 

170 

169 

1&9 

168 

168 

168 

167 

167 

166 

166 

166 

165 

165 

164 

164 

164 

163 

163 

163 

162 

162 

162 

161 

161 

160 

160 

160 

159 

159 

159 

158 

158 

158 

157 

157 

167 

166 

156 

156 

155 

156 

154 

154 

154 

153 

163 

163 

152 

152 

152 

152 

151 

151 

151 

150 


996751 
996735 
996720 
996704 
996688 
996673 
996657 
996641 
996625 
996610 
996594 
9.996578 
9965G2 
996546 
996530 
996514 
996498 
996482 
996465 
996449 
996433 
9.996417 
996400 
996384 
996368 
996361 
996335 
996318 
996302 
996286 
996269 
9.996252 
996236 
996219 
996202 
996186 
996168 
996151 
996134 
996117 
996100 
.996083 
996066 
996049 
996032 
996015 
995998 
996980 
995963 
995946 
995928 
9.995911 
995894 
995876 
995869 
996841 
995823 
996806 
995788 
995771 
996753 


2.6 
2.6 
2.6 
2.6 
2.6 


6 

6 

6 

6 

6 

7 

7 

7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.8 

2.8 

2.8 


Sine. 


2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 


Taiig: 


mrw 


9.089144 
090187 
091228 
092266 
093302 
094336 
095367 
096395 
097422 
098446 
099468 
9.100487 
101504 
102519 
103532 
104542 
106550 
106556 
107559 
108560 
109559 
9.110556 
111551 
112643 
113533 
114521 
115507 
116491 
117472 
118462 
119429 
9.120404 
121377 
122348 
123317 
124284 
126249 
126211 
127172 
128130 
129087 
1.130041 
130994 
131944 
132893 
133839 
134784 
136726 
136667 
137605 
138642 
.139476 
140409 
141340 
142269 
143196 
144121 
145044 
146966 
146885 
147803 
Cotang. 


174 

173 

173 

173 

172 

172 

171 

171 

171 

170 

170 

169 

169 

169 

168 

168 

168 

167 

167 

166 

166 

166 

165 

165 

165 

164 

164 

164 

163 

163 

162 

162 

162 

161 

161 

161 

160 

160 

160 

159 

159 

159 

168 

158 

168 

167 

157 

157 

156 

155 

156 

156 

165 

156 

154 

154 

154 

153 

153 

163 


Ootang.  I'N.  sine.  N.  con. 


10.910856 
909813 
908772 
907734 ! 
906698 
905664  i 
9046331 
903605  j 
902578  j 
901554  I 
900532  I 

10.899513 i 
898496 ! 
897481  I 
896468 
895458  I 
8944501 
893444  j 
892441 


12187 
12216 
12245 
12274 
12302 
12331 
12360 
12389 
12418 
12447 
12476 
12504 


99256 
99251 
99248 
99244 
99240 
99237 
99233 
99230 
99226 
99222 
99219 
99215 


12591 
12620 


12678 


12706  99189 


8914401 1 12735 
890441  1 112764 
10.889444  !j  12793  99178 
88844911 12822  99175 
887457  I  i 12851 
886467 
885479 

884493 II 12937199160 
883509  ij  12966 


12633  99211 
12562  99208 


99204 
99200 


12649  99197 


99193 


99186 
99182 


99171 


12880  99167 
12905  99163 


882528 
881548 
880571 
10.879596 
878623 
877652 
876683 


99156 
!  12995  99152 


13024 
13053 
13081 


99148 
99144 
99141 


13110  99137 
13139  99133 


13168  99129 
875716  j  13197  99125 
874751 '113226  99122 
873789  i!  13264)99118 
872828  1 1 13283199114 
871870  :ll3G12  99110 
870913  |!  13341  99106 
10.86995911 13370  99102 
99098 
99094 
867107  I  i  13456199091 
866161  1113485 


869.006  jilSG 99 

868056  113427 


865216! 
864274 
8633c!3 


13514 


13543  99079 
13572  99075 


862396 
861458 
10.860524 
859591 
868()60 
857731 
856804 
855879 
864956 

864034  j 13860 
8531 15 ''.13889 
862197  ii  13917 


13773 
13802 


Tang. 


?9087 
99083 


13600  99071 
13629199067 
13658  99063 
13687  99059 
1371C  99055 
13744,99051 


99047 
)9043 


13831  99039 
99035 
39031 
M9027 


N.  cos.  N.eine. 


82  Degrees. 


Log.  Sines  and  Tanj^ents.    (8°)    Natural  Sinca. 


29 


Bino. 


9.143555 
144453 
145349 
146243 
147136 
148026 
148915 
149802 
150686 
151569 
152451 

9.153330 
154208 
155083 
155957 
156830 
157700 
158569 
159435 
160301 
161164 

9.162025 
162885 
163743 
164600 
165454 
166307 
167159 
168008 
168856 
169702 

9.170547 
171389 
172230 
173070 
173908 
174744 
175578 
176411 
177242 
178072 

9.178900 
179726 
180551 
181374 
182196 
183016 
183834 
184651 
185466 
186280 

9.187092 
187903 
188712 
189519 
190325 
191130 
191933 
192734 
193534 
194332 
I  Cosine. 


D.  10' 


150 

149 

149 

149 

148 

148 

148 

147 

147 

147 

147 

146 

146 

146 

145 

145 

145 

144 

144 

144 

144 

143 

143 

143 

142 

142 

142 

142 

141 

141 

141 

140 

140 

140 

140 

139 

139 

139 

139 

138 

138 

138 

137 

137 

137 

137 

136 

136 

136 

136 

135 

135 

135 

135 

134 

134 

134 

134 

133 

133 


Cosine. 


9.996753 
995735 
995717 
995699 
995681 
995664 
995646 
995628 
996610 
995591 
995573 
9.995555 
995537 
995519 
995501 
995482 
995464 
995446 
995427 
995409 
995390 
9.995372 
995353 
995334 
995316 
996297 
995278 
995260 
995241 
996222 
995203 
995184 
995165 
995146 
995127 
995108 
995089 
995070 
995051 
995032 
995013 
9.994993 
994974 
994955 
994935 
994916 
994896 
994877 
994867 
994838 
994818 
.994798 
994779 
994759 
994739 
994719 
994700 
994680 
994660 
994640 
994620 
Sine. 


D.  10" 


3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.2 
3.2 
3.2 
3.2 
3.2 
8.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3,2 
3.2 
3.2 
3.3 
3.3 
3.3 
3.3 
3.3 


3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 


Tang. 


9.147803 

148718 
149632 
150644 
151464 
152363 
153269 
154174 
155077 
155978 
156877 
9.167775 
158671 
169666 
160467 
161347 
162236 
163123 
164008 
164892 
165774 
9.166654 
167532 
168409 
169284 
170157 
171029 
171899 
172767 
173634 
174499 

. 175362 
176224 
177084 
177942 
178799 
179655 
18060S 
181360 
182211 
183069 

. 183907 
184752 
185597 
186439 
187280 
188120 
188968 
189794 
190629 
191462 

. 192294 
193124 
193953 
194780 
196606 
196430 
197253 
198074 
198894 
199713 


D.  10'   Cotang.  |N.  sine.  N.  cos, 


153 

152 
152 
152 
151 
151 
151 
160 
160 
150 
150 
149 
149 
149 
148 
148 
148 
148 
147 
147 
147 
146 
146 
146 
145 
145 
146 
145 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
141 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 
137 
137 
137 
137 
136 


Co  tang. 


10.852197 
851282 
850368 
849456 
848546 
847637 
846731 
845826 
844923 
844022 
843123 

10.842225 
841329 
840435 
839543 
838653 
837764 
836877 
835992 
836108 
834226 


13917 

13946 

13975 

14004 

14033 

14061 

14090 

14119 

14148 

14177 

14205 

14234 

14263 

14292 

14320 

14349 

14378 

1440 

14436 

14464 

14493 


10.8333461 114522 


832468 
831691 
830716 
829843 
828971 
828101 
827233 
826366 
825501 

10.824638 
823776 
822916 
822058 
821201 
820345 
819492 
818640 
817789 
816941 

10.816093 
815248 
814403 
813561 
812720 
811880 
811042 
810206 
809371 i 
808538 

10.8077061 
806876 
806047  I 
806220  I 
804394 1 
803570  i 
802747 
801926  I 
8011061 
800287 


14551 
14580 
14608 
14637 
14666 
14695 
14723 
14752 
14781 
14810 
14838 
14867 
14896 
14925 
14954 
14982 
16011 
16040 
15069 
15097 
15126 
15155 
i 15184 
16212 
15241 
16270 
15299 
15327 
16356 
15385 


Tang. 


15442 
15471 
15600 
16629 
15557 
16586 
16615 
15643 


99027 
99023 
99019 
99015 
99011 
99006 
99002 


98994 
98990 


98982 
98978 
98973 
98969 
98965 
98961 
98957 
98953 


98944 
98940 


98931 
98927 


98919 
98914 
98910 
98906 
98902 
98897 
98893 


98876 
98871 
98867 
98863 
98858 
98854 
98849 
98845 
98841 


98827 
98823 
98818 
98814 
98809 


15414  98806 


98800 
98796 
98791 
98787 
98782 
98778 
98773 
98769 
N.  cos.  N.siDe 


81  Degrees. 


30 


Log.  Sines  aud  Tangents.    (9°)    Natural  Sines. 


TABLE  n. 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
IP) 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
BO 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


9.194332 
195129 
1959-25 
196719 
197511 
198302 
199091 
199879 
200666 
201451 
202234 

9.203017 
203797 
204577 
205354 
206131 
208906 
207679 
203452 
209222 
209992 

9.210760 
211626 
212291 
213055 
213818 
214679 
215338 
216097 
216854 
217609 

9.218363 
219116 
219868 
220618 
221367 
222116 
222861 
223606 
224349 
225092 

9.225833 
226573 
227311 
228048 
228784 
229518 
230252 
230984 
231714 
232444 

9.233172 
233899 
234625 
235349 
236073 
236795 
237515 
238235 
238953 
239670 


D.  ny 


Ck>8ine. 


133 

133 

132 

182 

132 

132 

131 

131 

131 

131 

130 

130 

130 

130 

129 

129 

129 

129 

128 

128 

128 

128 

127 

127 

127 

127 

127 

126 

126 

126 

126 

126 

125 

125 

125 

125 

124 

124 

124 

124 

123 

123 

123 

123 

123 

122 

122 

122 

122 

122 

121 

121 

121 

121 

120 

120 

120 

120 

120 

119 


Cosine. 

9.994620 
994600 
994580 
994560 
994540 
994519 
994499 
994479 
994459 
994438 
994418 
9.994397 
994377 
994357 
994336 
994316 
994295 
994274 
994254 
994233 
994212 
9.994191 
994171 
994150 
994129 
994108 
994087 
9940(36 
994045 
994024 
994003 
9.993981 
993960 
993939 
993918 
993896 
993875 
993854 
993832 
993811 
993789 
.993768 
993746 
993725 
993703 
993681 
993660 
993638 
993616 
993594 
993572 
.993550 
994528 
993506 
993484 
993462 
993440 
993418 
993396 
993374 
993351 


D.  It)' 


Sine. 


3.3 
3.3 
3.3 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3,4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3,5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.6 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.7 
3.7 
3.7 
3.7 
3-7 
3.7 


Taai;. 

3.199713 

200529 
201345 
202169 
202971 
203782 
204592 
205400 
206207 
207013 
207817 

).  208619 
209420 
210220 
211018 
211815 
212611 
213405 
214198 
214989 
215780 

).  216568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 
223606 

1,224382 
225156 
225929 
226700 
227471 
228239 
229007 
229773 
230589 
231302 

'.232066 
232826 
233586 
234345 
235103 
235859 
236614 
237368 
238120 
238872 

.239622 
240371 
241118 
241865 
242610 
243354 
2440{}7 
244839 
245579 
246319 

CoUmj;. 


136 
136 
136 
135 
136 
135 
136 
134 
134 
134 
184 
133 
133 
133 
133 
133 
132 
132 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
130 
129 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
126 
126 
126 
125 
125 
126 
124 
124 
124 
124 
124 
123 
123 


Cotang. 


N.  mne.lN.  cos 


10.800287 
799471 
798666 
797841 
797029 
796218 
796408 
794600 
793793 
792987 
792183 

10.791381 
790580 
789780  i 
788982 
788185 
787389 
786596 
785802 
735011 
784220 

10.783432 
782644 
781858 
781074 
780290 
779608 
778728 
777948 
777170 
776394 

10.775618 
774844 
774071 
773300 
772529 
771761 
770993 
770227  1 1 
769461  ! 


16643 

15672 

15701 

15730 

16758 

1578 

15816 

16846 

15873 

16902 

16931 

16959 


98769 
98764 
98760 
98766 
98751 
98746 
98741 
98737 
98732 
98728 
98723 
98718 


15988  98714 


16017 
16046 
16074 
16103 
16132 
16160 
16189 
16218 
16246 
16275 
16304 
16333 
16361 
16390 
16419 
16447 
16476 
16605 
16533 
16562 
13591 
16620 
16648 
16677 
16706 


98709 
98704 
98700 
98695 
98690 
98686 
98681 
98676 
98671 
98667 
98662 
98667 
98652 
98648 
98643 
98638 
98633 
98629 
98624 
98619 
98614 
98609 
98604 
98600 
98595 


10.767935! 
767174! 
766414  I 
765655 
764897  I 
764141 j 
763386  1 
762632 
761880  I 
761128 

10.760378 
759629 
758882 
758136 
757390 
756646 
755903 
756161 
764421 
763681 


16734  98590 
16763  98585 
16792  9S580 
16820,98576 


16849 
16878 
16906 
16935 
16964 
16992 
17021 
17050 


198570 
98565 
98661 
98556 
98551 
98546 
98541 
98536 


17078198531 
17107  98526 


Tang. 


98621 
98616 
98611 
98506 
98501 
98496 
98491 
98486 
98481 
N.  COS.  N.sine, 


17136 
17164 
17193 
17222 
17250 
17279 
17308 
17336 
17365 


80  T).vaveK. 


TAFLE  II. 


Log.  Sinc«  and  Tar.gcnt.s.    (10«-')    Kjjturjil  Sines. 


.31 


8 
9 
10 
11 
I'i 
13 
14 
15 
16 
17 
18 
19 
20 

22 
23 
24 
25 
2o 
2/ 

I  SO 
j3i 
i  32 
j  33 
j34 
I  35 
I  3;» 
!  3/ 
I  38 
I  39 
I  40 
41 
I  42 

i  "^^ 
4i 

4t) 
47 
48 
4y 
50 
51 
52 
53 
54 
55 
5b 
67 
58 
69 
60 


Sine. 

K 239670 
240386 
241101 
241814 
242526 
243237 

-  243947 
244655 
245363 
246069 
246775 

). 247478 
248181 
248883 
249583 
250282 
250980 
251677 
252373 
253067 
253761 

).  254463 
255144 
255834 
256523 
257211 
257898 
258683 
259268 
259951 
260633 

>.  26 1314 
261994 
262b73 
263351 
264027 
264703 
265377 
266051 
266723 
267395 

1.268065 
2t>8  /34 
269402 
2/0069 
270735 
271400 
272064 
272726 
273388 
274049 

1.274708 
276367 
276024 
276681 
2^77337 
277991 
278644 
279297 
279948 
280599 

C().«inf'. 


D.  10"  Cosiu 


119 
119 
119 
119 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
115 
115 
115 
115 
115 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 
112 
112 
112 
112 
111 
111 
1.11 
HI 
111 
111 
110 
110 
110 
110 
110 
110 
109 
109 
109 
109 
109 
109 
108 


1.993351 
993329 
993307 
993285 
993262 
993240 
993217 
993195 
993172 
993149 
993127 

•.993104 
993031 
993059 
993036 
993013 
992990 
992967 
992944 
992921 
992898 

.992875 
992852 
992829 
992806 
992783 
992759 
992736 
992713 
992690 
992666 

.992643 
992619 
992596 
992672 
992549 
992525 
992601 
992478 
992454 
992430 

.992406 
992382 
992359 
992335 
992311 
992287 
992263 
992239 
992214 
992190 

.992166 
992142 
992117 
992093 
992059 
992044 
992020 
991996 
991971 
991947 


D.  10" 

3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.8 


8 

8 

8 

8 

8 

8 

8 

8 

8 

3.8 

3.8 

3.8 

3.8 

3.8 

3.8 

3.8 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

3.9 

,9 

,9 

,0 

,0 

,0 

,0 


3 
3 

4 

4 

4 

4 

4.0 

4.0 

4.0 


4.0 
4.1 


Tanir. 

9.246319 
247057 
247794 
248530 
249264 
249998 
260730 
261461 
252191 
252920 
263648 

9.254374 
265100 
255824 
256547 
257269 
257990 
258710 
269429 
260146 
260863 

9.261578 
262292 
263005 
263717 
264428 
265138 
265847 
266555 
267261 
267967 

9.268671 
269375 
270077 
270779 
271479 
272178 
272876 
273573 
274269 
274964 

9.275668 
276361 
277043 
277734 
278424 
279113 
279801 
280488 
281174 
281858 

9.282642 
283225 
283907 
284588 
285268 
285947 
286<>24 
287301 
287977 
288662 


Co  tang. 


D.  1(»"|  CoUm/.     |.N..«ine.|N.  M*, 


123 
128 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
121 
120 
120 
120 
120 
120 
120 
119 
119 
119 
119 
119 
118 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
116 
116 
116 
116 
115 
115 
114 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 


il0. 753681 
752943 
752206 
751470 
750736 
750002 
749270 
748539 
747809 
747080 
74(i352 

10.745626 
744900 
744176 
743453 
742731 
742010 


1736598481 
17393  98476 
17422.98471 
17451  984{)6 
17479  98461 
17508  98455 
1753798450 
17565  98445 
17594  98440 
17623  98435 
17651 198430 
17680  98425 


741290 
740571 
739854 
739137 

10.738422 
737708, 
736995  i; 
736283 , 
735572  |! 
7348621; 
734153  ] 
733445  I 
73273911 
732033'! 

10.731329' 
730625 
729923 
729221 
728521  ii 
727822 
727124 
726427 
725731 
725036  jj 

10.7243421 
723649  I 
722957" 


17708 
17737 
17766 
17794 
17823 
17852 
17880 
17909 
17937 
17966 
17995 
18023 
18052 
18081 
18109 
18138 
18166 
18195 
18224 
18252 
18281 
18309 


98420 
98414 
98409 
98404 
98399 
9831*4 
98389 
98383 
98378 
98373 
98368 
9b  36  2 
98367 
98362 
98347 
98341 
98336 
98331 
98326 
98320 
98316 
98310 


18338  98304 
18367 198299 
18395198294 


18424 
18452 
18481 


98288 
96283 
98277 


18509  98272 
18538  98267 


79  Dojfreos. 


25 


:« 


Log.  Sines  and  Tangents.    (11°)    Natural  Sines. 


TABLE  II. 


blue. 


9.280599 
281248 
281897 
282544 
283190 
283836 
284480 
285124 
285766 
286408 
287048 

9.287687 
288326 
288964 
289600 
290236 
290870 
291504 
292187 
292768 
293399 

9.294029 
294658 
296286 
295913 
296539 
297164 
297788 
298412 
299034 
299655 

9.800276 
300895 
301514 
302132 
302748 
303364 
303979 
304593 
305207 
305819 

9.306430 
307041 
307650 
308259 
308867 
309474 
310080 
310686 
311289 
311893 

(9.312495 
313097 
313698 
314297 
314897 
315495 
316092 
316689 
317284 
317879 
Cosine. 


D.  10' 


108 
103 
108 
108 
108 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
105 
105 
105 
105 
105 
105 
104 
104 
104 
104 
104 
104 
104 
103 
103 
103 
103 
103 
103 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
99 
99 
99 


D.  10 " 


1.991947 
991922 
991897 
991873 
991848 
991823 
991799 
991774 
991749 
991724 
991699 

1.991674 
991649 
991624 
991599 
991574 
991549 
991524 
991498 
991473 
991448 

1.991422 
991397 
991372 
991346 
991821 
991295 
991270 
991244 
991218 
991193 

1.991167 
991141 
991115 
991090 
991064 
991038 
991012 
990988 
990960 
990934 

'  990908 
990882 
990855 
990829 
990803 
990777 
990750 
990724 
990697 
990671 

1.990644 
990618 
990591 
990565 
990638 
990611 
990485 
990458 
990431 
990404 
Sine. 


4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 


3 

3 

3 

3 

3 

3 

3 

3 

4.3 

4.3 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4 

4.5 

4.5 

4.5 

4.5 


.288662 
289326 
289999 
290671 
291342 
292013 
292682 
293350 
294017 
294684 
295349 

.296013 
296677 
297339 
298001 
298662 
299322 
299980 
300638 
301295 
301951 

.302607 
303261 
303914 
304567 
305218 
305869 
308519 
307168 
307815 
308463 

.309109 
309754 
310398 
311042 
311685 
312327 
312967 
313608 
314247 
314885 
9.315523 
316159 
316795 
317430 
318064 
318697 
319329 
319961 
320592 
321222 
321851 
322479 
323106 
323733 
324368 
324983 
325607 
326231 
326853 
327475 


Co  tan  R. 
Degrt^s. 


112 
112 
112 
112 
112 
111 
111 
111 
111 
111 
111 
111 
110 
110 
110 
110 
110 
110 
109 
109 
109 
109 
109 
109 
109 
108 
108 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
lOJ 
105 
105 
105 
105 
105 
106 
105 
104 
104 
104 
104 
104 
104 
104 
104 


ix)t.ang.   (N.  sine.  N.  cos 


10.711348 
710674 
710001 
709329 
708658 
707987 
707318 
706650 
705983 
705316 
704651 

10.703987 
703323 
702661 
701999 
701338 
700678 
700020 
699362 
698705 
698049 

10-697393 
696739 
696086 
696433 
694782 
694131 
693481 
692832 
692185 
691537 

10-690891 
690246 
689802 
688958 
688315 
687673 
687033 


19081 
19109 
19138 
19167 
19195 
19224 
19252 
19281 
19309 
19338 
19366 
19396 
19423 
19452 
19481 
19509 
19638 
19566 
19595 
19623 
19662 
19680 
19709 
19737 
19766 
197y4 
19828 
19861 
19880 
19908 
19937 
19966 
19994 
20022 
20061 
20079 
20108 
20136 
6863921120166 


20193 
20222 
20260 
20279 
20307 
120336 
20364 
20393 
20421 
20460 


685753 
6861161 

10-684477! 
683841 
683205 ' 
682570 
681936 
681303 
680671 i 

680^)39  i 

679408 'j  20478 
678778 i  1 20507 

10.678149^120535 
6776211  20563 
676894  I !  20692 
676267  i  1 20(i20 
675642  1 1 20649 
675017  j 20677 
674393!!  20706 
673769  1 1 20734 
673147  j 20763 
6725261120791 


Tang. 


98163 
98157 
98152 
98146 
98140 
98135 
98129 
98124 
98118 
98112 
98107 
98101 
98096 
98090 
98084 
98079 
)73 
98067 
98061 
98056 
98060 
98044 
98039 
98033 
98027 
98021 
98016 
98010 
98004 


979981 31 


97992 
97987 
97981 
97975 
97969 


97963  1  25 


97958 

97952 

97946 

97940 

97934 

97928 

97922 

97916117 

97910  10 


7905 
97899 
97893 
97887 
97881 
97876 
97869 
97863 
97867 
97851 
97846 
97839 
97833 
97827 
97821 
97815 


N.  COS.  IV.><in«'. 


TABLE  II. 


Log.  Bines  and  Tangents.    (12°)    Natural  Sines. 


33 


N.sine.iN.  cos, 


2079197815 
2082097809 
20848  97803 
20877  97797 
20905  97791 
20933  97784 
20962  97778 
2099097772 
21019  97766 
21047  97760 
21076  97754 
2110497748 
21132  97742 
2116197735 
•21189  97729 
21218  97723 
21246  97717 

121275  97711 
21303|97705 
2133197698 
21360,97692 
21388  97686 
2141797680 
21445  97673 
2147497667 
21502  97661 
21530  97655 
21559  97648 
2168797642 
2161697636 
21644|97630 
2167297623 
2170197617 
21729  97611 
21758J97604 
21786:97598 
2181497692 
21843197686 
21871  97579 
21899  97573 
21928  97566 
21956  97560 
2198597653 
22013  97547 

I  22041  97541 


22070 
22098 
22126 


97534 

97528 
621 


22156^7515 
22183  97508 
2221297602 
|22240'97496 
122268197489 
22297197483 
22325  ©7476 
22363  97470 
2238297463 
2241097457 
22438  97450 
22467  97444 
22495  97437 


I  N.  cos.lN.8ine 


77  Degree*, 


34 


Log,  Sines  and  Tangento.    (13°)    Natural  Sines. 


TABLE  n. 


Sine. 

9,352088 
352635 
353181 
353726 
354271 
354815 
355358 
355901 
356443 
356984 
357524 

9.368064 
358603 
359141 
359678 
360216 
3&0752 
361287 
361822 
362356 


D.  10' 


9.363422 
363964 
364485 
366016 
365546 
866075 
366604 
367131 
367659 
3681 85 

9.368711 
369236 
369761 
370286 
370808 
371330 
371852 
372373 
372894 
373414 

9.373933 
374462 
374970 
375487 
376003 
376519 
377035 
377649 
378063 
S78577 

9.379089 
379601 J 
380113 
380624 
381134 
381643 
382162 
382661 
383168 
383676 
Cosine.  ' 


Cosine. 


9,988724 
9^695 
988666 
988636 
988607 
988678 
988548 
988619 
988489 
988460 
98iW30 

9.988401 
988371 
988342 
988312 
968282 
988252 
988223 
988193 
988163 
988133 

».  988103 
988073 
988043 
988013 
987983 
987953 
987922 
987892 
987862 
987832 

9.987801 
987771 
987740 
987710 
987679 
987649 
987618 
987588 
9&7667 
987526 

9.987496 
987466 
987434 
987408 
987372 
987341 
987310 
987279 
987248 
987217 

9.987186 
987156 
987124 
987092 
987061 
987030 
986998 
986967 
986936 


D.  10' 


4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
4.9 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
5.0 
6.0 
5.0 
5.0 
6,0 
6.0 
5.0 
5.0 
5.1 
6.1 
6,1 
6.1 
6.1 
6.1 
6.1 
5.1 
5.1 


&.2 
6,2 
6.2 
6.2 
6.2 
6.2 
6.2 
5.2 
5.2 
6.2 
5.2 
6.2 
5.2 
5.2 
5.2 
6.2 


Tang. 

.363364 
363940 
364515 
365090 
365664 
366237 
366810 
367382 
367953 
368624 


9.369663 
370232 
370799 
371367 
371933 
372499 
378064 
373629 
374193 
374766 

9.375319 
376881 
376442 
377003 
377563 
378122 
378681 
379239 
379797 
380354 
380910 
381466 
382020 
382575 
383129 
383682 
384234 
384786 
385337 


9.386438 
386987 
387536 
388084 
388631 
389178 
389724 
390270 
390816 
391360 

1.391903 
392447 
392989 
393531 
394073 
394614 
396154 
396694 
396233 
396771 
Cotanp. 


D.  10' 


96.0 
95.9 
95.8 
95.7 
96,6 
95,4 
95,3 
95.2 
95.1 
95.0 
94.9 
94.8 
94.6 
94.6 
94.4 
94.3 
94.2 
94.1 
94.0 
93.9 
93.8 
93.7 
93.5 
93.4 
93.3 
93.2 
93,1 
93.0 
92.9 
92.8 
92.7 
92.6 
y2  6 
92,4 
92,3 
92,2 
93.1 
92.0 
91,9 
91,8 
91.7 
91.5 
91,4 
91.3 
91.2 
91.1 
91.0 
90.9 
90.8 
90.7 
90.6 
90.5 
90.4 
90.3 
90.2 
90.1 
90.0 
89.9 
89.8 
89.7 


.  Cotang.  I  N. sine 


Tang. 


!  22863197851 
!  22892197345 
973:^8 
97331 
97325 
97318 
311 
97304 
97298 
97291 
97284 
23176  97278 
97271 
97264 
97257 
23288  97251 
23316  97244 


23797 


23769  97134 


97127 


23825  97120 
97118 
9710t> 
97100 
97093 


97072 
97066 
24079  97  06;S 
97051 
24136  97044 
24164  97037 
24192  97030 


N.  COS.  N  sine.  '• 


76  Degrees. 


36 


Log^  Sines  and  Tangents.    (15°)    Natural  Sines. 


0 
1 

2 
3 
4 
6 
6 
7 
8 
9 

la 
11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
21 
28 
29 
30 
31 
32 
33 
34 
35 
36 
87 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
64 
55 
56 
51 
58 
59 
60 


Sine. 

J. 412996 
413467 
413938 
414408 
414878 
415347 
415815 
416283 
416751 
417217 
417684 

). 418150 
418615 
419079 
419544 
420007 
420470 
420933 
421395 
421857 
422318 

). 422778 
423238 
4236^7 
424166 
424616, 
425073 
425530 
425987 
426443 
426899 

). 427354 
427809 
428263 
428717 
429 L70 
429623 
430a76 
430527 
430978 
431429 

>. 431879 
432329 
432778, 
433226 
433675 
434122 
434569 
435016 
435462 
435908 

11.436353 
436798, 
437242 
437686 
438129 
438572 
439014 
439456 
4C9897 
440338 
CoiJine, 


D.   10" I  Ijosinc. 

k.  984944 
984910 
984876 
984842 
984808 
984774 
984740 
984706 
984672 
984637 
984603 
.984569 
984535 
984500 
984466 
984432 
984397 
984363 
984328 
984294 
984259 
.984224 
984190 
984155 
984120 
984085 
984050 
984016 
983981 
983946 
983911 
.983875 
983840 
983805 
9837'^0 
983755 
983700 
983664 
983629 
983594 
983558 
.983523 
983487 
983452 
983416 
983381 
983345 
983309 
983273 
983238 
983202 
.983166 
983130 
983094 
983058 
983022 


78.5 
78.4 
78.3 
78.3 
78.2 
78.1 
78.0 
77.9 
77.8 
77.7 
77.6 
77.5 
77.4 
77.3 
77.3 
77.2 
77.1 
77.0 
76.9 
76.8 
76.7 
76.7 
76.6 
76.5 
76.4 
76.3 
76.2 
76.1: 
76.0 
76.0 
75.9 
75.8 
75.7 
75.0 
75.5 
75.4 
75.3 
75.2 
75.2 
75.1 
75..  0, 
74.9 
74.9 
74.8 
74.7 
74  6 
74.5 
74.4 
74.4 
74.3 
74.2 
74.1 
74. ft 
74.0 
73.9 
73.8 
73.7 
73.6 
73.6 
73.51 


982950 
982914 
982878 
982842 


Sine. 


JX  W\      Tang. 


5.7 
6.7 
5.7 
6.7 
5.7 
5.7 
6.7 
5.7 
5.7 
5.7 
5.7 
6.7 
6.7 
5.7 
6.7 
5.8 
6.8 
5. .8 
5.8 
5v8 
5.8 
5.8 
5.8 
5.8 
6.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
6.9 
5.9 
5.9 
5.9 
5.9 
5.9 
6.9 
5.9 
5.9 
6.9 
6.9 
6.9 
5.9 
5.9 
5.9 
6.9 
6.0 
6.0 
6.0 
6.0 
6.0 

6.  a 

6.0 
6.0 
6.0 
6.0 
6.0 
6.0 


9.428062 
428557 
429062 
429561 
430070 
430573 
431075 
431577 
432079 
432580 
433080 

9.433580 
434080 
434579 
435078 
435576 
436073 
436670 
437067 
437563 
438059 

9.438554 
439048 
439543 
440036 
440529 
441022 
441614 
442006 
442497 
442988 

9.443479 
443968 
444458 
444947 
445.435 
44a923 
446411 
446898 
447384 
447870 
1.448356 
448841 
449326 
449810 
46Q294 
460777 
45L260 
451743 
452226 
452706 
.453187 
453668 
454148 
464628 
465107 
465586 
456064 
456542 
457019 
457496 
Cotanjj. 


a  iiy; 

84.2 
84.1 
84.0 
83.9 
83.8 
83.8 

83.7 

83.6 
83.5 
83.4 
83.3 
83.2 
83.2 
83.1 
83.0 
82.9 
82.8 
82.8 
82.7 
82.6 
82.6 
82.4 
82.3 
82.3 
82.2 
82.1 
82.0 
81.9 
81.9 
81.8 
81.7 
81.6 
81.6 
81.5 
81.4 
81.3 
81.2 
81.2 
81.1 
81.0 
80.9 
SO. 9 
80.8 
80.7 
8ft.  6 
80.6 
80.5 
80..  4 
80.3 
80.2 
80.2 
80. 1 
80.0 
79.9 

9.9 
79.8 
79.7 

9.6 
79.6 
79.5 


Lotuup;.   N.  sine.  N.  COS. 

96593 
96585 
96578 
96570 
96562 
96665 
96547 
96640 
96532 
96524 


10.571948!  25882 
571443  i  269 10 
570938  ':  2593 
670434'  2596 
6r>9930;i25994 
569427  1 126022 
668925!  126050 
5684231126079 
567921  126107 
667420  ,26135 


5669201126163  96517 


10.566420: '26191 
5669201:26219 
5654311126247 
564922  1126275 
5644241126303 
663927  1 126331 
5634301126359 
562933  I  26387 
562437  [26416 
561941  !  26443 

10.561446!!  26471 
560952  1 1 26500 
560457  1  26528 


26556 
S6584 
26612 
26640 
26668 
26696 


559964 
559471 
658978 
558486 
657994 
557503 

557012  1 1 26724 
10-.666521H  26762 
556032  1126780 
5555421126808 
555053!!  26836 
654566 
554077 
553589 
553102 
552616 
652130- 
0..  561644 


551159 


26864 
26892 
26920 
26948 
26976 
27004 
27032 
37060 


650674  127088  96261 


650190  127116 
549706  127144 
549223  127172 
5487401127200 


648267  ;  27228  96222 


547776  ij  27256 
547294  1 1 27284 
10.5468131;  27312 
546332 
545852 
545372 
644893 
544414 
643936 
543458 
542981 
542504 


Tang. 


96509 
96502 
96494 
96486 
96479 
96471 
96463 
96456 
96448 
96440 
96433 
96425 
96417 
96410 
96402 
96394 
96386 
96379 
96371 
96363 
96355 
96347 
96340 
96332 
96324 
96316 
96308 
96301 
96293 
96285 
96277 
96269 


96253 
96246 
96238 
96230 


96214 

96206 

961.98 

7340  96190 

27368  96182 

27396  96174 

27424  96166 

27452  96158 

27480  96150 

27508  96142 

27536  96134 


27564 


96126 


N.  cos.lN.sine. 


74  Degrees. 


TABLE  II. 


Log.  Sinefl  and  Taflgcnts.    (16°)    Natural  Sines. 


37 


Hine. 

).4t0338 
440778 
441218 
441658 
442096 
442636 
442973 
443410 
443847 
444284 
444720 

). 445165 
445590 
446026 
446469 
446893 
447326 
447759 
448191 
448623 
449054 

). 449486 
449915 
450346 
450776 
461204 
451632 
452060 
452488 
452916 
453342 

). 453768 
454194 
454619 
455044 
455469 
456893 
456316 
456739 
467162 
457584 

3  468006 
458427 
458848 
459268 
459688 
460108 
460527 
460946 
461364 
461782 

9.462199 
462616 
463032 
463448 
463864 
464279 
464694 
465108 
466522 
465935 


U.  10"     Coiiiio.     D.  lu 


Cosine. 


73,4 
73.3 
73.2 
73-1 
73.1 
73.0 
72.9 
72.8 
72.7 
72.7 
72.6 
72.6 
72.4 
72.3 
72.3 
72.2 
72.1 
72.0 
72.0 
71.9 
71.8 
71.7 
71.6 
71.6 
71.6 
71.4 
71.3 
71.3 
71.2 
71.1 
71.0 
71.0 
70.9 
70.8 
70.7 
70.7 
70.6 
70.5 
70.4 
70.4 
70.3 
70.2 
70.1 
70.1 
70.0 
69.9 
69.8 
69.8 
69.7 
69.6 
69.5 
69.5 
69.4 
69.3 
69.3 
69.2 
69.1 
69.0 
69.0 
68.9 


9.982842 
982805 
982769 
982733 
982696 
982660 
982624 
982587 
982551 
982514 
982477 
982441 
982404 
982367 
982331 
982294 
982257 
982220 
982183 
982146 
982109 

9.982072 
982036 
981998 
981961 
981924 
981886 
981849 
981812 
981774 
981737 

9.981699 
981662 
981625 
981587 
981649 
981512 
981474 
981436 
981399 
981361 

9.981323 
981285 
981247 
981209 
981171 
981133 
981096 
981057 
981019 
980^)81 
,980942 
980904 
980866 
980827 
980789 
980750 
980712 
980S73 
980635 
980596 
Sino. 


6.0 
6.0 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 


6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.3 


6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 


Tanj;. 

9.457496 
467973 
468449 
458925 
459400 
469875 
460349 
460823 
461297 
461770 
462242 
462714 
463186 
463658 
464129 
464699 
466069 
466539 
466008 
466476 
466946 
467413 
467880 
468347 
468814 
469280 
469746 
470211 
470676 
471141 
471605 
472068 
472632 
472995 
473457 
473919 
474381 
474842 
475303 
476763 
476223 

9  476683 
477142 
477601 
478059 
478517 
478976 
479432 
479889 
480345 
480801 
481257 
481712 
482167 
482621 
483076 
483529 
483982 
484436 
484887 
485339 


1>.  lo'   Coian;:.  ;  N.ainc.  N.  coa 


79.4 
79.3 
79.3 
79  2 
79.1 
79.0 
79.0 
78.9 


10.542504!  27564 
5420271' 27692 
641651  i  27620 


Cotan<r. 


78 

78 

78 

78 

78 

78 

78 

78 

78.3 

78.2 

78.1 

78.0 

78.0 

77.9 

77.8 

77.8 

77.7 

77.6 

77.5 

77-5 

77.4 

77.3 

77.3 

77.2 

77.1 

77.1 

77.0 


76.9 
76.8 
76.7 
76.7 
76.6 
76.5 
76.5 
76.4 
76.3 
76.3 
76.2 
76.1 
76.1 
76.0 
76.9 
75.9 
76.8 


75.3 


541075 

540600 

540126 

539651 

539177 

538703 

538230 ! 

537758 
10.637286 

536814 

536342 

535871 

635401 

534931 

534461 

538992 

533524 

533055 
10.532587 

632120 

531663 

531186 

530720 

530254 

529789 

529324 

528859 

628395 
10.527932 

627468 

627005 

526643  ! 

526.081  I 

625619 1 

5261581 

524697 

524237 

523777 
10  523317 

522858 

522399 

521941 

521483 . 

5210251 128847 

520568  112887  6 

520111  j  28903 

5196651128931 

619199  jl 28959 
10.518743  1  28987 

518288!  29015 

517833  {129042 

517379  H2907O 

516925  I  i  29098 

516471  1 129126 

516018  1 29154 

515665  '29182 

615113  |l2920iJ 

5146611129247 


27648 
27676 
27704 
27731 
27759 
27787 
27815 
27843 
27871 
27899 
27927 
27955 
27983 
28011 
28039 
28067 
28095 
28123 
28150 
28178 
28206 
28234 
28262 
28290 
28318 
28346 
28374 
28402 
28429 
28467 
28485 
28513 
28541 
28569 
28597 
28625 
28652 
28680 
28708 
28736 
28764 
128792 
128820  95 


Tanp. 


N.  COP.  N.8in 


96126 
96118 
96110 
96102 
96094 
96086 
96078 
96070 
.%062 
96064 
96046 
96037 
96029 
96021 
96013 
96005 
d5997 
95989 
96981 
95972 
96964 
95956 
95948 
95940 
96931 
95923 
95915 
95907 
95898 
95890 
95882 
95874 
95865 
95857 
95849 
96841 
95832 
95824 
95816 
S6807 
95799 
96791 
95782 
95774 
95766 
95757 
95749 
95740 
95732 
95724 
95716 
95707 
95698 
95690 
96681 
96673 
95664 
95656 
95647 
95639 

966;;o 


73  D<'a;rce8. 


38 


Log.  Sines  and  Tangents.    (17°)    Natural  Sines.  TABLE  II. 


! Sine.     p.  10" 


7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
28 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
88 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
62 
53 
64 
55 
56 
57 
68 
69 
60 


.465935 
466348 
466761 
467173 
467585 
467996 
468407 
468817 
469227 
469037 
470046 
9.470455 
470803 
471271 
471679 
472086 
472492 
472898 
473304 
473710 
474115 
9.474519 
474923 
475327 
475730 
476133 
476536 
476938 
477340 
477741 
478142 
■478542 
478942 
479342 
479741 
480140 
480539 
480937 
481334 
481731 
482128 
9.482525 
482921 
483316 
483712 
484107 
484501 
484895 
485289 
485682 
486075 
,486407 
486860 
487251 
487643 
488034 
488424 
488814 
489204 
489593 
489982 


68.8 

68.8 

08.7 

68.6 

68,5 

68.5 

68.4 

68.3 

68.3 

68.2 

68.1 

68.0 

68.0 

67.9 

67.8 

67.8 

67.7 

67.6 

67.6 

67.5 

67.4 

67.4 

67.3 

67.2 

67.2 

67.1 

67.0 

66.9 

66.9 

60.8 

66.7 

60.7 

66.6 

66.5 

66.5 

66.4 

66.3 

66.3 

66.2 

66.1 

66.1 

66.0 

65.9 

65,9 

65.8 

65.7 

65.7 

65,6 

65.5 

65.5 

65.4 

66.3 

65.3 

65.2 

65.1 

65.1 

66.0 

65.0 

64.9 

64.8 


Cosine. 


Cosine. 

9.980596 
980558 
980519 
960480 
980442 
980403 
980364 
980325 
980286 
980247 
980208 
.980109 
980130 
980091 
980052 
980012 
979973 
979934 
979895 
979855 
979816 
9.979776 
979737 
979697 
979658 
979618 
979579 
979539 
979499 
979459 
979420 
.979380 
979340 
979300 
979260 
979220 
979180 
979140 
979100 
979059 
979019 
9.978979 
978939 
978898 
978858 
978817 
978777 
978736 
978696 
978655 
978616 
,978574 
978533 
978493 
978452 
978411 
978370 
978329 
978288 
978247 
978206 


D.  10" 


Sine. 


6.4 

6.4 

6.5 

6.5 

6.6 

6.5 

6.5 

6.5 

6.5 

6.6 

6.5 

6.5 

6.6 

6.5 

6.6 

6.5 

6.5 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 

6.6 


6.6 
6.6 
6.6 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6,8 
6.8 
6.8 
6.8 
6.8 
6.8 


9.485339 
485791 
480242 
486693 
487143 
487593 
488043 
488492 
488941 
489390 
489838 
3.490286 
490733 
491180 
491627 
492073 
492519 
492965 
493410 
493854 
494299 

9.494743 
495186 
495630 
496073 
496515 
496957 
497399 
497841 
468282 
498722 

9.499163 
499603 
600042 
500481 
500920 
601359 
601797 
502235 
502672 
603109 
503546 
503982 
504418 
504854 
605289 
505724 
606169 
606593 
507027 
507460 
507893 
508326 
608769 
509191 
609622 
510054 
510486 
510916 
611346 
611776 


D.  10" 


Co  tang. 


75.3 

75.2 

76.1 

76.1 

76.0 

74.9 

74.9 

74.8 

74.7 

74.7 

74.6 

74.6 

74.5 

74.4 

74,4 

74.3 

74,3 

74.2 

74.1 

74.0 

74.0 

74.0 

73.9 

73.8 

73.7 

73.7 

73.6 

73,6 

73.5 

73.4 

73.4 

73.3 

73.3 

73.2 

73.1 

73.1 

73.0 

73.0 

72.9 

72.8 

72,8 

72,7 

72.7 

72.6 

72.5 

72.6 

72.4 

72.4 

72.3 

72.2 

72.2 

72,1 

72,1 

72.0 

71.9 

71.9 

71,8 

71.8 

71.7 

71.6 


Cotang. 


10.614661 
614209 
613768 
613307 
612867 
512407 
611957 
611508 
611059 
510610 
510162 

10.509714 
609267 
608820 
608373 
607927 
607481 
507035 
506590 
606146 
505701 

10.505257 
604814 
604370 
603927 
503486 
603043 
602601 
602159 
501718 
501278 

10,500837 
600397 
499958 
499519 
499080 
498641 
498203 
497765 
497328 
496891 

10.496454 
496018 
495582 
495146 
494711 
494276 
493841 
493407 
492973 
492540 

10.492107; 
491674 
491241 
490809 
490378 
489946 
489516 
489084 
488054 
488224 


N.  sine.fN.  cos, 


29237 
29265 
29293 
29321 
29348 
29376 
29404 
29432 
29460 
29487 


29543 
29571 
29599 
29626 
29654 


29682  95493 


29710 
29737 
29766 
29793 
29821 


96630 

95622 
95613 
95605 
95696 
95588 
95679 
95671 
95562 
95654 


29616  96645 


95536 
96528 
95619 
95511 
95502 


95485 
95476 
95467 
96459 
95450 


29849  95441 


29876 
29904 
29932 
29960 


29987  95398 
30015  95389 
30043  95380 


30071 
30098 
30126 
30154 
30182 
30209 
30237 
30265 
30292 
30320 
30348 


95372 
S5363 
95354 
95345 
95337 
95328 
96319 
95310 
95301 
95293 
95284 


30376  95275 


30403 
30431 
30459 
30486 
30514 
30542 
30570 
30697 
30025 
30G53 


30763 
30791 


Tang.   11  N.  com 


95433 
95424 
95415 
96407 


95260 
95257 
95248 
95240 
95231 
95222 
95213 
95204 
96196 
95186 


30080  95177 
30708  95168 
30730  95159 


95150 
195142 


30819  95133 
30840  95124 
30874  95115 
3090^  95100 


60 
59 
58 
57 
66 
65 

64 

53 

52 

51 

50 

49 

48 

47 

46 

46 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 
3  I 

32 

31 

30 

29 

28 

27 

20 

25 

24 

23 

22 

21 

20 

19 

18  I 

17 

10 

15 

14 

13 

12 

u 

10 
9 
8 
7 
0 
5 
4 
3 

1 
0 


72  Degrees. 


TABLE  II. 


Log.  Since  and  Tangeut*.    (18®)    ^'atu^al  Sines. 


39 


Sint'. 

9.489982 
490371 
490759 
491147 
491535 
491922 
492308 
492695 
493081 
493466 
493861 

9.494236 
494621 
495005 
495388 
495772 
496154 
496537 
496919 
497301 
497682 

9.498034 
498444 
498825 
499204 
499584 
499963 
500342 
50U721 
501099 
501476 

9.501864 
602231 
502607 
502984 
603360 
503736 
504110 
604485 
604860 
605234 

9.505608 
505981 
603354 
506727 
507099 
507471 
507843 
508214 
508585 
508956 

9.509326 
509696 
510066 
610434 
510803 
511172 
511540 
611907 
512276 
612642 
Cosine. 


D.  W 


64.8 
64.8 
64.7 
64.6 
64.6 
64.6 
64.4 
64.4 
64,3 
64.2 
64.2 
64.1 
64.1 
64.0 
63.9 
63.9 
63.8 
63.7 
63.7 
63.6 
63.6 
63.6 
63.4 
63.4 
63.3 
63.2 
63.2 
63.1 
63.1 
63.0 
62.9 
62.9 
62.8 
62.8 
62.7 
62.6 
62.6 
62.5 
62.6 
62.4 
62.3 
62.3 
62.2 
62.2 
62.1 
62.0 
62.0 
61.9 
61,9 
61.8 
61.8 
61.7 
61.6 
61.6 
61.6 
61.5 
61.4 
61.3 
61.3 
61.2 


Cosine. 

1.978206 
978165 
978124 
978083 
978042 
978001 
977959 
977918 
977877 
977835 
977794 

1.977752 
977711 
977669 
977628 
977588 
977544 
977603 
977461 
977419 
977377 

1.977335 
977293 
977251 
977209 
977167 
977125 
977083 
977041 
976999 
970967 
6914 
976872 
976830 
976787 
976745 
976702 
976660 
976617 
976574 
976532 

•976489 
976446 
976404 
976361 
976318 
976275 
976232 
976189 
976146 
976103 

.976060 
976017 
975974 
975930 
975887 
975844 
975800 
975757 
976714 
975670 


9.9' 


Sine. 


P.  10' 

6.8 
6.8 
6.8 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 


7.0 
7.0 


7.1 
7.1 
7.1 
7.1 

7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 


JTuuu:. 

9.511776 
612206 
512636 
613064 
513493 
513921 
614349 
514777 
515204 
515631 
516057 
,516484 
516910 
517336 
517761 
518185 
518610 
519034 
519458 
519882 
520305 

9.520728 
621161 
521673 
521993 
622417 
522838 
523259 
523680 
524100 
524520 
524939 
525359 
526778 
626197 
526615 
527033 
527451 
527868 
628285 
528702 
629119 
629535 
529950 
530366 
530781 
631196 
531611 
532025 
532439 
532863 

9.533266 
633679 
634092 
634604 
534916 
636328 
635739 
636150 
536561 
636972 
Cotang. 


D.  10" 


71 

71 

71 

71 

71 

71 

71,3 

71,2 

71,2 

71.1 

71.0 

71.0 

70.9 

70,9 

70.8 

70,8 

70.7 

70.6 

70.6 

70.5 

70.6 

70.4 

70.3 

70.3 

70.3 

70.2 

70.2 

70.1 

70.1 

70,0 

69,9 


69,8 
69.7 
69,7 
69,6 
69,6 
69,5 
69,6 
69,4 
69,3 
69.3 
69.3 
69.2 
69,1 
69.1 
69  0 
69.0 
68.9 
68.9 
68  8 
68.8 


Cotang.  i  N.  sine 


10.488224 
487794 
487365 


486507 
486079 
486651 
485223 
484796 
484369 
483943 

10.483516 
483090 
482665 ! 
482239 
481815  i 
481390  I 
480966 { 
480542 
480118 
479696  ' 

10.479272 
478849 
478427 
478005 
477583 
477162 
476741 
476320 
475900 
475480 

10.475061 
474641 
474222 
473803 
473385 
472967 
472549 
472132 
471715 
471298 

10.470881 
470465 
470050 
469634  ! 
469219  I 
468804  I 
468389  I 
467975  } 
467561 
467147  I 

10.466734 
466321 
465908 
465496 
465084 
464672 
464261 
463850  I 
463439 
463028 
"Twig. 


130902 
130929 
130957 
1 30985 
31012 
31040 
1 31068 
131095 
131123 
31161 
131178 
131206 
131233 
131261 
31289 
J31316 
31344 
31372 
31399 
31427 
31454 
31482 
31510 
31537 
31565 
31593 
31620 
31648]! 
31675 
31703 
31730 


N.  COS 


95106 
95097 
96088 
96079 
96070 
95061 
95052 
95043 
96033 
95024 
95016 
95006 
94997 
94988 
94979 
94970 
94961 
94952 
94943 
94933 
94924 
94915 
94906 
94897 
94888 
94878 
94869 
94860 
94851 
94842 
94832 
3175894823 
31786  94814 
31813  948U5  I  27 


60 
59 

58 
57 
56 
55 
64 
63 
62 
51 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 
31 
30 
29 


94571 
94661 
94552 
N.  COS.  N.6ine, 


71  Degrees. 


49 


Log.  Sines  and  Tangents.    (13°)    Natural  Sines. 


TABLE  II. 


0 

1 

2 
S 

4 
6 
6 

7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
36 
36 
37 
88 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


3.512642 
61300D 
613375 
613741 
514107 
514472 
514837 
515202 
515566 
515930 
516294 

}. 516657 
517020 
517382 
517745 
618107 
618468 
618829 
619190 
619551 
519911 

). 520271 
520631 
520990 
621349 
521707 
622066 
622424 
522781 
523138 
523495 

). 523852 
524208 
624564 
524920 
626276 
525 'j30 
525984 
520339 
52e693 
527046 

I. 627400 
527753 
628106 
528458 
528810 
529161 
629513 
629864 
530215 
630565 

•.530916 
531265 
531614 
531963 
632312 
532661 
533009 
533357 
533704 
534062 

Cosine. 


61.2 
61.1 
61.1 
61.0 
60.9 
60.9 
60.8 
60.8 
60.7 
60.7 
60.6 
60.5 
60.5 
60.4 
60.4 
60.3 
60.3 
00.2 
00.1 
60.1 
60.0 
60.0 
59.9 
59.9 
59.8 


59 

59 

59 

69 

59 

59 

59 

69 

59 

59.3 

59.2 

59.1 

59.1 

59.0 

59.0 

58.9 

58.9 

68.8 

58.8 

58.7 

58  7 

58.6 

58.6 

58.5 

68 

58 

58 

58 

58 

58 

58 

58.1 

68.0 

58.0 

57.9 


OOlSiUO. 

.975670 
9756-27 
975583 
975539 
975496 
975452 
975408 
975365 
975321 
976277 
975233 
.975189 
975145 
975101 
976057 
975013 
974969 
974925 
974880 
974836 
974792 
.974748 
974703 
974659 
974614 
974570 
974525 
974481 
974436 
974391 
974347 
.974302 
974257 
974212 
974167 
974122 
974077 
974032 
973987 
973942 
973897 
.973852 
973807 
973761 
973716 
973671 
973625 
973580 
973536 
973489 
973444 
.973398 
973362 
973307 
973201 
973215 
973169 
973124 
973078 
973032 
972986 
"Shie^ 


7.3 
7.3 
7.3 
7.3 
7.3 
7.3 
7.3 
7.3 


7.3 
7.3 
7.3 
7.3 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.6 
7.5 
7.6 
7.6 
7.5 
7.5 
7.6 
7.6 
7.5 
7.5 
7.5 
7.5 
7.6 
7.5 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 


7.7 


9.536972 
637382 
537792 
538202 
638611 
539020 
539429 
539837 
540245 
540653 
541061 

9.541468 
541875 
542281 
542688 
543094 
643499 
643J>05 
644310 
644715 
645119 

9.545524 
545928 
546331 
646735 
647138 
547540 
547943 
548345 
648747 
649149 

9.540650 
549951 
550352 
560752 
551162 
661552 
551962 
552351 j 
562750 
553149 

9.553548 
553946 
554344 
554741 
656139 
665536 
555933 
556329 
556725 
557121 

1.557517 
667913 
558308 
658702 
659097 
659491 
659885 
660279 
560673 
561066 

"CotangT 


68.4 

68.3 

68.8 

68.2 

68.2 

68.1 

68.1 

68.0 

68.0 

67.9 

67.9 

67.8 

67.8 

67.7 

67.7 

67.6 

67.6 

67.5 

67.5 

67.4 

67.4 

67.3 

67.3 

67.2 

67,2 

67.1 

67.1 

67.0 

67.0 

66. 

66. 

66. 

66.8 

66.7 

66.7 

66.6 

66.6 

66.6 

66.5 

66.5 

66.4 

66.4 

66.3 

66.3 

66.2 

66.2 

66.1 

66.1 

66.0 

66.0 

65.9 

66.9 

66.9 

65.8 

65.8 

65.7 

65.7 

66.6 

65.6 

65.6 


Cotan);. 


ijiN'.  Binc.|;>i.  COS. I 


32557 
32584 
32612 


10.463028 

462618  j 

462208  j 

4617981 1 32639 

461389!  32667 

460980'  32694 

460571  I  32722 

460163  I 

459765  i 

459347  I 

458939  i 
10.4585321 

458125 j 

467719  I 

457312  I 

456900  I 

450501  ! 

456095 ! 

455690 1 

455285 

454881  I 
10.454476! 

464072  j 

453669  I 

453265  ! 

452862  I 

452460 ! 

452057 

451655 

451253 

450861 
LO. 450450 

460049 ! 

449648 

449248 

448848 

448448 

448048 


32749 
32777 
32804 
32832 
32859 
32887 
32914 
32942 
32969 
32997 
33024 
33051 
33079 
33106 
33134 
.33161 
33189 
33216 
33244 
33271 
33298 
33326 
33353 
33381 
33408 
33436 
33463 
33490 
33518 
33645 
33573 


447649  1 1 33600 
447250 1 1 33627 
446851 1 1 33665 
33682 


10.446452 
446054 
445656 
445259 
444861 
444464 


83710 
33737 
33764 
33792 
33819 


444067  1133846 
443671  I  [33874 
4432761133901 

33929 

3395b 

33983 

34011 

34038 

34065 

34093 

84120 

3414 

4393271134175 
438934 1134202 
Tanjr.   !lN.  cos.  N.sine, 


94552 

94542 

94533 

94523 

94514 

94504 

94495 

94485 

94476 

94466 

94457 

94447 

94438 

94428 

94418 

94409 

94399 

94390 

94380 

94370 

94361 

94351 

94342 

94332 

94322 

94313 

94503 

94293 

94284 

94274 

94264 

94254 

94245 

94235 

94225 

94215 

94206 

94196 

94186 

94176 

94167 

94167 

94147 

94137 

94127 

94118 

94108 

94098 

94088 

94078 

94068 

94058 

94049 

94039 

94029 

94019 

94009 

93999 

93989 

93979 

93969 


70  Degre'js. 


TABLE  II. 


Log.  i;iire«  and  TanReiitn.    (20<=>)     >aluraJ  Siuoh. 


41 


Sine.   p.  W 


534052 
534399 
534745 
535092 
535438 
535783 
536129 
55G474 
536818 
537163 
537507 
537851 
538194 
538538 
538880 
539223 
539565 
539907 
540249 
540590 
540931 
541272 
541618 
541953 
542293 
542632 
542971 
543310 
543649 
543987 
544325 
9,544663 
545000 
545338 
545674 
546011 
546347 
546683 
547019 
547354 
547689 

1.548024 
548359 
548693 
549027 
549360 
649693 
550026 
550359 
550692 
551024 

1.551356 
551687 
552018 
552349 
552680 
653010 
553341 
553670 
554000 
554329 
Cosine. 


57.8 
57.7 
67.7 
67-7 
67.6 
57.6 
57.5 
57.4 
67.4 
57.3 
57.3 
57.2 
57.2 
67.1 
57.1 
67.0 
57.0 
56.9 
56.9 
66.8 
56.8 
56.7 
66.7 
56.6 
56.6 
56.5 
56.6 
58.4 
66.4 
56.3 
56.3 
56.2 
66.2 
66.1 
56.1 
66.0 
56.0 
55.9 
55.9 
56.8 
55.8 
55.7 
65.7 
65.6 
65.6 
55.6 
65.5 
55.4 
55.4 
55.3 
55.3 
56.2 
55.2 
55.2 
55.1 
55.1 
55.0 
55.0 
54.9 
54.9 


Oosiui*. 

). 972986 
972940 
972894 
972848 
972802 
972755 
97270:1 
972663 
972617 
972570 
972524 

). 972478 
972431 
972385 
972338 
972291 
972245 
972198 
972161 
972105 
972058 

>. 972011 
971964 
971917 
971870 
971823 
971776 
971729 
971682 
971635 
971688 

(.971540 
971493 
971446 
971398 
971351 
971303 
971256 
971208 
971161 
971113 

>.  971 066 
971018 
970970 
970922 
970874 
970827 
970779 
970731 
970683 
970635 

1.970586 
970538 
970490 
970442 
970394 
970345 
970297 
970249 
970200 
970162 


D.  10' 


Sine. 


7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7,9 
7,9 
7.9 
7.9 
7.9 
7.9 
7.9 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8,0 
8.0 
8.0 
8.0 
8,0 
8,0 
8,0 
8.0 
8,1 
8.1 
8.1 


JTunj^ 

9.561066 
661459 
561851 
662244 
662636 
563028 
563419 
663811 
664202 
564592 
564983 

9,565373 
565763 
566163 
666542 
566932 
667320 
567709 
668098 
568486 
568873 

9.569261 
569648 
670035 
570422 
570809 
571196 
571581 
571967 
672362 
572738 

9.573123 
573507 
573892 
574276 
574660 
576044 
675427 
676810 
576193 
576676 

9.576968 
577341 
677723 
678104 
578486 
578867 
579248 
679629 
580009 
580389 

9.580769 
681149 
581528 
581907 
582286 
582665 
583043 
583422 
583800 
584177 
Cotanp. 


D.  10 


65.5 
65.4 
65.4 
65  3 
66.3 
65.8 
65.2 
65.2 
65.1 
65.1 
65.0 
65.0 
64.9 
64,9 
64.9 
64.8 
64.8 
64.7 
64.7 
64.6 
64.6 
64.6 
64,5 
64,5 
64,4 
64,4 
64.3 
64.3 
64,2 
64.2 
64.2 
64.1 
64.1 
64.0 
64.0 
63,9 
68.9 
63,9 
63.8 
63.8 
63.7 
63.7 
63,6 
63.6 
63.6 
63,5 
63,6 
63,4 
63.4 
63,4 
63,3 
63,3 
63.2 
63,2 
63,2 
63,1 
63.1 
63.0 
63,0 
62.9 


t'otang.   >.  Kine.  N,  cos. 


10.438934 
438541 
438149 
437756 
437364 
436972 
436581 
436189 
436798 
435408 
435017 

10.434627 
434237 
433847 
433458 
433068 
432680 
482291 
431902 
431514 
431127 

1 a. 430739 
430362 
429965 
429678 
429191 
428806 
428419 
428033 
427648 
427262 

10.426877 
426493 
426108 
425724 
425340 
424956 
424573 
424190 
423807 
423424 

10.423041 
422659 
422277 
421896 
421614 
421183 
420762 
420371 
419991 
419611 

10.419231 
418851 
418472 
418093 
417714 
417336 
416957 
416578 
416200 
415823 


Tanp;. 


,  3420-: 
1 34229 
i  34257 
134284 
134311 
34339 
34366 
34393 
34421 
34448 
34475 
34503 
84530 
34657 
34584 
34612 
84639 
34666 
34694 
34721 
84748 
84776 
34803 
34830 
84857 
84884 
34912 
349G9 
34966 
34993 
35021 
35048 
85076 
35102 
85130 
8515 
35184 
35211 
35289 
35266 
35293 
35320 
3584 
35375 
35402 
35429 
35456 
35484 
35511 
35538 
35565 
3559i; 
35619 
3564'i 
85674 
35701 
3572b 
35755 
3578i. 
85810 
35837 


93077 
93667 
93657 
93647 
93637 
93626 
93616 
93606 
93596 
93585 
i)3576 
93565 
93555 
93644 
93534 
93524 
93514 
93503 
93493 
93483 
98472 
J3462 
93452 
93441 
93431 
93420 
98410 
93400 
93f,fc9 
93379 
98368 
93368 
i  N.  COP.  N.sine. 


60 
I  59 
68 
57 
56 
55 
54 
63 
L2 
ol 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
86 
85 
84 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
:.0 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


09  Degrcps. 


42 


Log.  Sines  and  Tangents.    (21")    Natural  Sines. 


TABLE  IL 


Sine. 


9.554329 
554658 
554987 
555315 
555(J43 
555971 
556299 
556626 
556953 
557280 
557()06 

9.557932 
558258 
558583 
558909 
559234 
55y558 
559883 
560207 
560531 
560855 

9.561178 
561501 
561824- 
662146 
562468 
562790 
563112 
563433 
563755 
664075 

9.564396 
564716 
665036 
565356 
565676 
565995 
566314 
666632 
566951 
567269 

9.567587 
567904 
568222 
568539 
568856 
569172 
569488 
569804 
670120 
570435 

9.570751 
571066 
671380 
571695 
572009 
572323 
672636 
672950 
673263 
673575 
Cosine. 


D.  10"|  Cosine. 


64.8 
64.8 
64.7 
54.7 
54.6 
64.6 
54.5 
54.5 
54.4 
54.4 
54.3 
54.3 
54.3 
54.2 
54.2 
54.1 
54,1 
54.0 
54.0 
53.9 
53.9 
53.8 
53.8 
63.7 
53.7 
53.6 


53 

53 

53 

53 

53 

63 

53 

53 

63 

53 

53.1 

53.1 

53,1 

63.0 

53.0 

52,9 

52.9 

52.8 

52.8 

62.8 

52.7 

52.7 

52.6 

52.6 

52.5 

52.6 

52.4 

62.4 

52.3 

52.3 

52.3 

52.2 

52.2 

52.1 


9.970152 
970103 
970055 
970006 
969957 
969909 
969860 
9698  U 
969762 
969714 
969665 

9.969616 
969567 
969518 
969469 
969420 
969370 
969321 
969272 
969223 
969173 

(9.969124 
969075 
969025 
968976 
968926 
968877 
968827 
968777 
968728 
968678 
968628 
968578 
968528 
968479 
968429 
968379 
968329 
968278 
968228 
968178 
968128 
968078 
968027 
967977 
967927 
967876 
957826 
967775 
967725 
967674 

9.967624 
967573 
967622 
967471 
967421 
967370 
967319 
967268 
967217 
967166 


D.  1U"|  Tang. 


8 

8 

8.2 

8.2 

8,2 

8.2 

8.2 

8.2 

8.2 

8.2 


2 

2 
2 

2 
2 
2 
3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 


9.584177 
584555 
684932 
585309 
686686 
686062 
686439 
586815 
687190 
587566 
587941 

9.588316 
688691 
5890d6 
689440 
689814 
590188 
590562 
690935 
691308 
591681 

9.692054 
59242*6 
592798 
593170 
593542 
593914 
594285 
594656 
595027 
595398 
595768 
596138 
696508 
596878 
597247 
697616 
597985 
698364 
598722 
699091 

9.599459 
599827 
600194 
600562 
600929 
601296 
601662 
602029 
602395 
602761 

1.603127 
603493 
603858 
604223 

.  604588 
604953 
605317 
605682 
60)046 
606410 

Cotang. 


D.  10"|  Cotang.  I  N  .sine.lN.  cos. 


62.9 

62.9 

62.8 

62 

62.7 

62.7 

62 

62 

62 

62 

62 

62 

62 

62 

62 

62 

62.3 

62.2 

62.2 

62.2 

62.1 

62,1 

62.0 

62.0 

61.9 

61.9 

61.8 

61.8 

61.8 

61.7 

61.7 

61.7 

61.6 

61.6 

61.6 

61.6 

61.6 

61.5 

61.4 

61.4 

61.3 

61.3 

61.3 

61.2 

61.2 

61.1 

61.1 

61.1 

61,0 

61.0 

61.0 

60.9 

60.9 

60.9 

60,8 

60.8 

60.7 

60.7 

60.7 

60.6 


10.415823 
415445 
416068 
414691 
414314 
.  413938 
413561 
413185 
412810 
412434 
412059 

10.411684 
411309 
410934 
410560 
410186 
409812 
409438 
409065 
408692 
408319 

10.407946 
407574 
407202 
406829 
406458 
406086 
405715 
405344 
404973 
404602 

10.404232 
403862 
403492 
403122 
402753 
402384 
402015 
401646 
401278 
400909 

10.400541 
400173 
399806 
399438 
399071 
398704 
398338 
397971 
397605 
397239  '■' 

10.396873, 
396507  I 
396142 : 
395777 : 
395412 
395047  '■ 
394683 ; 
394318  : 
393954 
393590 ■ 


36837  93358 
36864193348 
35891 193337 


35918 
35945 
36973 
36000 
36027 
36054 
36081 
36108 
36135 
36162 


36190  93222 


36217 


93211 


i  36244  93201 


36271 


93190 


36298  93180 


36325 


93169 


36461 
36488 
36515 


36542  93084 
!  36569  93074 
ij  36596  93063 
1 136623  93052 


36731 

36758 
36785 


j  1 368 12  92978 
1 136839  92967 


!  136867 


36894  92946 


! 36921 


93327 
93316 
93306 
93295 
93286 
93274 
93264 
93253 
93243 
93232 


36352  93159 
36379^3148 
36406  93137 
36434  93127 


93116 
93106 
93095  I 


36650  93042 
36677193031 
36704  93020 


93010 
92999 
92988 


92966 


92935 


Tang.   I N.  i 


136948  92926 
!  36975  92913 
137002  92902 
137029  92892 
i  37056  92881 
'  37083  92870 
137110  92859 
:  37 137  92849 
37164  92838 
3719192827 
:  37218  92816 
37245  92805 
37272  92794 
i  37299  92784 
137326  92773 
37353  92762 
37380  92751 
137407  92740 
j  37434  92729 
13746192718 


Log.  Sines  and  Tangents.    (22'')    Natural  Sines. 


43 


Sine. 


9,573675 


574200 
674512 
574824 
576136 
575447 
575758 
576069 
576379 
576689 

►.576999 
677300 
577618 
577927 
578236 
578545 
678853 
579162 
579470 
679777 

1.580085 
580392 
580699 
581005 
681312 
681618 
681924 
682229 
582636 
582840 

.583145 
683449 
583754 
684058 
584361 
584665 
584968 
686272 
685574 
585877 

.586179 
686482 
686783 
587085 
587386 
687688 
687989 
588289 
588690 
688890 

i.  6891 90 
589489 
589789 
590088 
E90387 
590686 
590984 
691282 
591680 
591878 
Cosine. 


D7W> 


52.1 
52.0 
52.0 
51.9 
51.9 
61.9 
51.8 
51.8 
51.7 
51.7 
51.6 
51.6 
51.6 
61.6 
51.5 
61.4 
51.4 
51.3 
51.3 
51.3 
51.2 


51.2 
51.1 
61.1 
51.1 
51.0 
51.0 
60.9 
60.9 
50.9 
50.8 
50.8 
60.7 
60,7 
50.6 
50.6 
60.6 
60.6 
50.5 
50.4 
60.4 
60.3 
60.3 
60.3 
50.2 
60.2 
60.1 
50.1 
50.1 
60.0 
50.0 
49.9 
49.9 
49,9 
49.8 
49.8 
49.7 
49.7 
49.7 
49.6 


Cosine. 


.967166 
967115 
967064 
967013 
966961 
966910 
966859 
966808 
966756 
966705 
966653 
.966602 
966550 
966499 
966447 
966395 
966344 
966292 
966240 
966188 
966136 
.966085 
966033 
965981 
965928 
966876 
965824 
965772 
965720 
965668 
965815 
.965563 
966511 
965458 
966406 
965353 
965301 
965248 
965195 
965143 
965090 
965037 
964984 
964931 
964879 
964826 
964773 
964719 
964666 
964613 
964560 
.964607 
964464 
964400 
964347 
964294 
964240 
964187 
964133 
964080 
964026 


Sine. 


D. 10"   Tan" 


8.5 
8,5 
8.5 

a,5 

8.5 
&.5 
8.6 
8.5 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 


). 606410 
606773 
607137 
607500 
607863 
608225 
608588 
608950 
609312 
609674 
610036 

). 610397 
610759 
611120 
611480 
611841 
612201 
612561 
612921 
613281 
613641 

>.  614000 
614369 
614718 
615077 
615435 
615793 
616161 
616509 
616867 
617224 

1.617582 
617939 
618295 
618652 
619008 
619364 
619721 
620076 
620432 
620787 

1.621142 
621497 
621852 
622207 
622561 
622915 
623269 
623623 
623976 
624330 

1.624683 
626036 
625388 
625741 
626093 
626445 
626797 
627149 
627601 
627852 

Co  tang. 
67  Degrees. 


D.  10' 


60.6 
60.6 
60.5 
60.6 
60.4 
60.4 
60.4 
60.3 
60.3 
60.3 
60.2 
60.2 
60.2 
60.1 
60.1 
60.1 
60.0 
60.0 
60.0 
59.9 
59.9 
59.8 
59.8 
59.8 
59.7 
59.7 
69.7 
59.6 
69.6 
59.6 
59.5 
59.5 
69.5 
69.4 
59.4 
59.4 
59.3 
59.3 
59.3 
59.2 
59.2 
59.2 
59.1 
59.1 
59.0 
59.0 
59.0 
58.9 
68.9 
68.9 
58.8 
58.8 
58.8 
58.7 
58.7 
58.7 
58.6 
58.6 
58.6 
58.5 


Cotang.     ;  N .  Hine.l N. 


10.393590 
393227 
392863 
392500 
392137 
391775 
391412 
391050 
390688 
390326 
389964 

10.389603 
389241 
388880 
388520 
388169 
387799 
387439 
387079 
386719 
386359 

10.386000 
385641 
385282 
384923 
384565 
384207 
383849 
383491 
383133 
382776 

10-382418 
382061 
381706 
381348 
380992 
380636 
380279 
379924 
379568 
379213 

10-378858 
378503 
378148 
377793 
377439 
371085 
376731 
376377 
376024 
375670 

10.375817 
374964 
374612 
374259 
373907 
373665 
373203 
372851 
372499 
372148 
f^iig." 


37461 
37488 
37515 
37542 
37569 
37695 
37622 
37649 
37676 
37703 
37730 
37757 
37784 


92718 
92707 
92697 
92686 
92675 
92664 
92653 
92642 
92631 
92620 
92609 
92598 
92587 


3781192676 


37838 
37865 
37892 
37919 
37946 
37973 
37999 
38026 
38053 
38080 
38107 
38134 
38161 
38188 


38241 
38268 
38295 
38822 
38349 
38376 
38403 
38430 


92565 
92554 
92543 
92532 
92521 
92510 
92499 
92488 
92477 
92466 
92456  j  36 


92444 
92432 
92421 


38215  92410 


92399 
92388 
92377 
92366 
92355 
92343 
92332 
92321 
38456  92310 
38483192299 
38510J92287 
38537  92276 
38564  92265 


38591 
38617 
38644 
38671 
38698 
38725 


92254 
92243 
92231 
92220 
92209 
92198 


38778  92175 
38805  92164 
38832  92152 
3885992141 
38886i92130 


38912 
38939 
38966 
38993 
39020 
39046 
39073 
N.  CO?. 


02119 
92107 
92096 
92085 
92073 
92062 
[92050 
>^Bine. 


35  I 

34 

03 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8  i 

7  : 

6 

5 

4  , 

3 

2 

1 

0 


44 


I>og.  Sine«  and  Tangents.     (23°)    Natural  Sines. 


TABLK  II. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
1() 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
I  31 
i  32 
33 
34 
85 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 

9.591878 
592176 
592473 
592770 
693067 
593303 
593659 
593955 
594251 
594547 
594S42 

9.595137 
595432 
595727 
598021 
596315 
696609 
596903 
597196 
597490 
597783 

9.598075 
598368 
598660 
598952 
599244 
599536 
599827 
600118 
600409 
600700 

9.600990 
601280 
601570 
601860 
602150 
602439 
602728 
603017 
603395 
603594 

9.603882 
604170 
604457 
604745 
605032 
605319 
605606 
605892 
606179 
60S465 

9.606751 
607030 
607322 
607607 
607892 
608177 
608461 
608745 
609029 
609313 


D  W- 


Cotinu 


49.6 
49.5 
49.6 
49.5 
49.4 
49.4 
49.3 
49.3 
49.8 
49.2 
49.2 
49.1 
49.1 
49.1 
49.0 
49.0 
48.9 
48.9 
48.9 
48.8 
48.8 
48.7 
48.7 
48.7 
48.6 
48.6 
48.6 
48.6 
48.6 
48.4 
48.4 
48.4 
48.3 
48.3 
48.2 
48.2 
48.2 
48.1 
48.1 
48.1 
48.0 
48.0 
47.9 
47.9 
47.9 
47  8 
47.8 
47.8 
47.7 
47.7 
47-6 
47.6 
47.6 
47-5 
47-6 
47.4 
47.4 
47.4 
47.3 
47.3 


Oosinc. 

1.964026 
963972 
963919 
963865 
963811 
963757 
963704 
963650 
963596 
963542 
963488 
.963434 
963379 
963325 
963271 
963217 
963163 
963108 
963054 
962999 
962945 
.962890 
962836 
962781 
962727 
962672 
962617 
962562 
962508 
962453 
962398 
.962343 
962288 
962233 
962178 
962123 
962037 
962012 
961957 
961902 
961846 
.961791 
961735 
961680 
961624 
961569 
961513 
961458 
961402 
961346 
961290 
.961235 
961179 
961123 
961037 
961011 
960955 
960899 
960843 
960786 
960730 
Sine. 


Tang. 

9.627852 
628203 
628564 
628906 
629255 
629606 
629956 
630306 
6306.56 
631005 
631355 

9.631704 
632053 
632401 
632750 
633098 
633447 
633795 
634143 
634490 
634838 
,635l8o 
635532 
635879 
636226 
636572 
636919 
637265 
637611 
637956 
638302 
638647 
638992 
639337 
639682 
640027 
640371 
640716 
641060 
641404 
641747 

9.642091 
642434 
642777 
643120 
643463 
643806 
644148 
644490 
644832 
645174 
.645516 
645857 
640199 
646540 
646881 
647222 
647662 
<347903 
648243 
648583 
Cotanii. 


^6  De^nvF. 


58.5 

58.5 

58.5 

58.4 

58.4 

68 

58 

58 

58 

58 

58 

58 

58.1 

68.1 

58.1 

58.0 

58.0 

58.0 

57.9 

57.9 

57.9 

57.8 

67.8 

67.8 

57.7 

67.7 

57.7 

57.7 

57.6 

57.6 

57.6 

57.5 

67.5 

57.5 

57.4 

67.4 

57.4 

67.3 

57.3 

67.3 

57.2 

57.2 

57.2 

57.2 

57.1 

57.1 

67.1 

57.0 

•57.0 

57.0 

66.9 

56.9 

66.9 

m.o 

56.8 
66.8 
56.8 
66.7 
56.7 
56.7 


Cotang.  I  N.  sine.  N.  COS. 


10.372148  139078  92050 


371797 '139100 
3714461  [39127 
371095!' 39153 
370746  1139180 


i  39207 
1 39234 
! 39260 


370394 
370044 
369694 
369344 
368996 

368646  1 39341 

10.368296  '39367 

367947  i 39394 

367599!  139421 


92039 
92028 
92016 
92005 
91994 
91982 
91971 


39287  91959 

39314  91948 

91936 

91925 

91914 

91902 

367250 II 39448  91891 

366902 1 1 39474 '91879 


366553  r  39501 
366205  ii  39528 
366857  j  139555 
365610!  1 39581 


3651621139608  91822 


10.364815 '139636 
364468 1 139661 
364121 ! 
363774 ! 
363428 ! 
363081  I 
362735  i 
362389 ' 
362044 1 
361698 ! 


39741 


39795 


39848 
39875 


91868 
91856 
91845 
91833 


91810 

91799 

39688  91787 

39715  91775 


91764 


39768  91752 


91741 


39822  91729 


91718 
91706 


10.361353  !i  39902  91694 
361008  II 39928191683 
360G63 'I  39955 19 1671 
360318  l|  39982  91660 


91648 
91636 
[40062  91625 
,40088  91613 


359973!  1 40008 
359629  II  40035 
359284 
358940 
358596!|40115|91601 
358253  ij  40141  bloiiO 
1 0. 357909  I  j  401 68  b  167  8 


357o66 ' 
867223  I 
356880 1 
356537  I 
366194!  14030) 


40195191566 
40221 191555 


40248 


191543 


40275  91531 
91519 


365862  1140328  9 16(j8 
36.5610  j  40355  91496 
355168  j 40381 


354826 
10.354484 
364143 
353801 
363460 
353119 
362778 
352438 
352097 
351757 
351417 


40408 
40434 


91484 
91472 
91461 


40461  91449 


40488 


40541 
40567 
40594 


40621  91378 


91366 
91355 
Tang,   ji  N.  cop.  N.siTic 


40647 
40674 


91437 


40514  91425 


91414 
91402 
91390 


60 

59 
58 
57 
66 
55 
54 
53 
52 
61 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
•■il 
fcO 
29 
•2b 
27 
26 
25 
'J  4 
23 
22 
21 
2U 
19 
18 
17 
16 
15 
14 
Ki 
12 

n 

10 
9 
8 
7 
6 
5 
4 

2 
1 
0 


Log.  Sines  and  Tangents.  (2-40)  Natural  Sines. 


45 


Sine. 


0 

9.609313 

609597 

2 

609880 

3 

610164 

4 

610447 

5 

610729 

6 

611012 

7 

611294 

8 

611576 

9 

611858 

10 

612140 

11 

9.612421 

12 

612702 

13 

612983 

14 

613264 

15 

613545 

16 

613825 

17 

614105 

18 

614385 

19 

614666 

20 

614944 

21 

9.615223 

22 

615502 

23 

615781 

24 

616060 

25 

616338 

26 

616616 

27 

616894 

28 

617172 

29 

617450 

30 

617727 

31 

9.618004 

32 

618281 

83 

618558 

34 

618834 

36 

619110 

36 

619386 

37 

619662 

38 

619938 

39 

620213 

40 

620488 

41 

9.620763 

42 

621038 

43 

621313 

44 

621587 

46 

621861 

46 

622135 

47 

622409 

48 

622682 

49 

622956 

50 

623229 

51 

9.623512 

52 

623774 

63 

624047 

54 

624319 

56 

624591 

56 

624863 

57 

626136 

68 

625406 

59 

625677 

60 

625948 

D.  10' 


47.3 
47.2 

47.2 
47.2 
47.1 
47.1 
47.0 
47.0 
47.0 
46.9 
46.9 
46.9 
46.8 
46.8 
46.7 
46.7 
46.7 
46.6 
46.6 
46.6 
46.6 
46.6 
46.5 
46.4 
46.4 
46.4 
46.3 
46.3 
46.2 
46.2 
46.2 
46.1 
46.1 
46.1 
46.0 
46.0 
46.0 
45.9 
45.9 
45.9 
45.8 
45.8 
45.7 
45.7 
46.7 
46.6 
46.6 
45.6 
46,5 
45.5 
46.6 
45.4 
45.4 
45.4 
45.3 
45.3 
46,3 
45.2 
46.2 
45.2 


Cosine. 

1.960730 
960674 
960618 
960561 
960505 
960448 
960392 
960336 
960279 
960222 
960165 

1.960109 
960052 
959996 
959938 
959882 
959825 
959768 
959711 
959664 
959596 

i.  959539 
969482 
959425 
959368 
959310 
959263 
969195 
969138 
959081 
969023 

.958966 


D.  ic/ 


958850 
958792 
958734 
958577 
958619 
958661 
958603 
958445 
958387 
958329 
958271 
958213 
958154 
958096 
968038 
957979 
957921 
957863 
967804 
957746 
967687 
957628 
957570 
957511 
957462 
957393 
957335 
957276 


9.4 

9.4 

9.4 

9.4 

9.4 

4 

4 

4 

4 

4 

4 

6 

6 

5 

5 


9.6 
9.6 
9.5 
9.5 
9.5 
9.5 
9.5 
9.5 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 


Tanft. 


9.648583 
648923 
649263 
649602 
649942 
660281 
650620 
660959 
651297 
651636 
651974 

9.652312 
652650 
652988 
653326 
653663 
654000 
654337 
654174 
665011 
666348 
655684 
656020 
666356 
656692 
657028 
657364 
667699 
658034 
658369 
658704 

9.659039 
659373 
659708 
660042 
660376 
660710 
661043 
661377 
661710 
662043 

9.662376 
662709 
663042 
663375 
663707 
664039 
664371 
664703 
665035 
666366 

3,666697 
666029 
666360 
666691 
667021 
667352 
667682 
6()8013 
668343 
668672 


D.  W 


66.6 
66.6 
66.6 
56  6 
66.5 
66.6 
59.6 
66.4 
56.4 
66.4 
66.3 
56.3 
56.3 
56.3 
56.2 
56.2 
66.2 
56.1 
56.1 
56.1 
66.1 
66.0 
56.0 
56.0 
56.9 
66.9 
55.9 
55.9 
65.8 
55.8 


Uotang. 


Cotang. 


65 

55 

65.6 

55.6 

55.6 

65.6 

55.5 

55 

55 

55 

56 

55 

55 

56 

56.3 

55.3 

55.2 

65.2 

55.2 

65.1 

55.1 

56.1 

56.1 

66.0 

55.0 

56.0 


10.351417 
351077 
350737 
350398 
350058 
349719 
349380 
349041 
348703 
348364 
348026 

10.347688 
347350 
347012 
346674 
346337 
346000 
345663 
345326 
344989 
344652 

10.344316 
343980 
343644 
343308 
342972 
342636 
342301 
341966 
341631 
341296 

10.340961 
340627  I 
340292  i 
339958 
339624 
339290 
338967 
338623 
338290 
337957 

10.337624 
337291 
336958 
336625 
336293 
336961 
336629 
335297 
334965 
334634 

10.334303 
333971 
333620 
333309 
332979 
332648 
332318 
331987 
331657 
331328 


N.  sine.  N.  cos. 


40674 
40700 
40727 
40753 
40780 
40806. 
40833 
40860 
40886 
40913 


40966 
40992 
41019 
41045 
41072 
41098 


41161 
41178 
41204 
41231 
41257 


91365 
91343 
91331 
91319 
91307 
91295 
91283 
91272 
91260 
91248 
91236 
91224 
91212 
91200 
91188 
91176 
91164 


41125  91162 


91140 
91128 
91116 
91104 
91092 


41284  91080 
4131091068 
41337  "91056 


Tang. 


41363 
41390 
41416 
41443 
41469 
41496 


41522  90972 


41549 
41675 


90960 
90948 


41602  90936 
41628  90924 


41655 
41681 
41707 
41734 
41760 
41787 
41813 
41840 
41866 


41919 
41946 
41972 
41998 
42024 


91044 
91032 
91020 
91008 
90996 
90984 


90911 
90899 
90887 
90875 
90863 
90861 
90839 
90826 
90814 


41892  90802 


90790 
90778 
90766 
90753 
90741 


4205190729 
42077  90717 
4210490704 
42130190692 
42166J90680 
42183  90668 
42209  90656 
42235  90643 
42262  90631 
N.  COS.  N. sine. 


65  Degrees. 


46 


Log.  Siucs  and  Tant^enu?.    (ti6  )     N^axiiral  Sines.  TAULE  II. 


Cotaug.     I  N  .siuc.  N 

10.331327! 
330998 
330Jei8  :' 
380339 
330009 
329(i80! 
329351;: 
32902311 
328694:; 

328037;; 
10.327709  I 

3273811 

327053;; 

32672611 

326398  ]> 

326071 ': 

325743 1 ; 

32541611 

325090!  I 

3247631; 
10.324436;! 

3241 10 j  I 

323784;; 

3234571; 

323131  i 

322806  ji 

322480  it 

32215411 

321829  li 

321504 
10.321179 

320854 

320529 

3202061! 

319880  i 

319556  I 

319232]! 

31890811 

318684!! 

31826011 
10.317937  i 

317613 

31729011 

316967;: 

316644/ 

316321  i! 

31599911 

3156761: 

3153541! 

31503211 
10.314710;; 

314388!; 

814066  i; 

313745!; 

313423;; 

3131021: 

312781 

31246a 

312139 

311818 


.Sina. 


D.  10" 


9.625948 

626219 

626490 

6267i)0 

627030 

627300 

627570 

627840 

628109 

628378 

628647 
9.628916 

629185 

629453 

629721 

629989 

630257 

630524 

630792 

631059 

631326 
9.631593 

631859 

632125 

632392 

632658 

632923 

633189 

633454 

6337191 1: 

633984 
9.634249 

634514 

634778 

635042 

635308 

635570 

635834 

636097 

636360 

636623 
9.636886 

637148 

637411 

637673 

637935 

638197 

638458 

638720 

638981 

639242 
9.639503 

639764 

640024 

640284 

640644 

640804 

641064 

641324 

641684 

641842 
Cosine.  I 


45, 
45, 
45, 
45. 
45, 
45, 
44. 
44, 
44, 
44, 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
44. 
4A. 
44. 
44. 
144. 


9.8 
9.8. 
9.& 
9,8 
9.8 
9.8 
9,9 
9.9 
9.9 
9.9 
9  9 


Cosine.  |0.  10' 

9,957276 

9572 17 

967158 

957099 

957040 

956981 

956921 

956862 

956803 

956744 

956684 
9.956625 

956566 

9565 J6 

956447 

956387 

956327 

956268 

956208 

956148 

956089 
9.956029 

955969 

956909 

955849 

955789 

956729 

955669 

955609 

955548 

955488 
9.955428 

955368 

955807 

955247 

955186 

955126 

9550G6 

9550U5 

95494-4 

964883 
9.954823 

954762 

964701 

954640 

954579 

964518 

954457 

954396 

954335 

954274 
9.954213 

954152 

964090 

954029 

953968 

953906 

953845 

953783 

953722 

953660 

Sin-'. 


9  9 
9.9 
9  9 
9  9 
9.9 
9.9 
9.9 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.1 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.3 


Tang. 

9.668673 
669002 
669332 
669661 
669991 
670320 
670649 
670977 
671306 
671634 
671963 

9.672291 
672619 
672947 
673274 
673602 
073929 
674257 
674584 
674910 
676237 

9.675664 
675890 
676216 
676543 
676859 
677194 
677520 
677846 
678171 
678496 

9.678821 
679146 
67947J 
„679795 
680120 
680444 
680768 
681092 
681416 
681740 

9.682063 
682387 
682710 
683083 
683356 
683679 
684001 
684324 
684646 
684968 

9.686290 
085612 
685934 
686256 
686577 
686898 
687219 
687640 
687861 
688182 
Co  tan  J. 
64  Dcgrwfl. 


D.  10" 


42262 
4228fc 
42816 
4-J841 
4236, 
4239-1 
424:.i0 
42446 
42478 
42499 
42525 
4255'..' 
42578 
42604 
42631 
42667 
42683 
42709 
4273() 
42762 
42788 
42815 
42841 
42867 
42894 
42920 
42946 
42972 
42999 
43025 
43051 
43077 
43104 
43130 
43166 
43182 
43209 
48235 
43261 
43287 
43313 
43340 
43366 
43392 
43418 
43445 
43471 
43497 
43528 
43549 
43575 
43602 
43G28 
43654 
43680 
43706 
43738 
43759 
43785 
43811 
43837 


Tane. 


90631 
^iHJlo 

J0594 

90669 
90557 
90545 
90532 
)0520 
90507 
90495 
90483 
90470 
90458 
90446 
90433 
90421 
90408 
90396 
90383 
90371 
90358 
90346 
90334 
90321 
9030y 
90296 
90284 
90271 
90259 
90246 
90233 
90221 
90208 
90196 
90188 
90171 
90158 
90146 
90188 
90120 
901 08 
90095 
90082 
90070 
90057 
90045 
90082 
90019 
90007 
89994 
89981 
89968 
89956 
89943 
89930 
89918 
89905 
89892 
89879 


Log.  Sine*  and  Tangents.  (26°)  Natural  Sines. 


47 


Sine.   D.  10"  Cosine. 


641842 
642101 
642360 
642618 
642877 
643135 
643393 
643660 
643908 
644166 
644423 
644680 
644936 
645193 
645450 
645706 
645962 
646218 
646474 
64G729 
646984 
647240 
647494 
647749 
648004 
648268 
648512 
648766 
649020 
649274 
649527 
649781 
650034 
650287 
650639 
650792 
651044 
651297 
651549 
651800 
652052 

9.652304 
652665 
652806 
653057 
653308 
653558 
653808 
654059 
654309 
654558 

9.654808 
665058 
655307 
655556 
655805 
656054 
656302 
656651 
656799 
657047 


43.1 
43.1 
43.1 
43.0 
43.0 
43.0 
43.0 
42.9 
42.9 
42.9 
42.8 
42.8 
42.8 
42.7 
42.7 
42.7 
42.6 
42.6 
42.6 
42.5 
42.5 
42.5 
42.4 
42.4 
42.4 
42.4 
4a.  3 
42.3 
42.3 
42.2 
42,2 
42.2 
42.2 
42.1 
42.1 
42.1 
42.0 
42.0 
42.0 
41.9 
41.9 
41.9 
41.8 
41.8 
41.8 
41.8 
41.7 
41.7 
41.7 
41.6 
41.6 
41.6 
41.6 
41.5 
41.5 
41.5 
41.4 
41.4 
41.4 
41.3 


Cosine,  j 


26 


9.953660 
953599 
953537 
963475 
953413 
953352 
953290 
953228 
958166 
953104 
963042 

9.952980 
952918 
952855 
952793 
962731 
952669 
952606 
952544 
952481 
952419 

9.952356 
962294 
952231 
952168 
962106 
952043 
951980 
951917 
951854 
951791 

9.951728 
951665 
951602 
951539 
961476 
961412 
961349 
961286 
951222 
951169 
951096 
951032 
960968 
950905 
950841 
950778 
950714 
950650 
950586 
950522 
950458 
950394 
950330 
950366 
950202 
960188 
960074 
950010 
949945 
949881 
"sine. 


D.  10"   Tang. 


10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.3 


10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10.5 

10.6 

10.6 

10.5 

10.5 

10.6 

10.5 

10.5 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.6 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 

10.7 


688182 
688502 
688823 
689143 
689463 
689783 
690103 
690423 
690742 
691062 
691381 
691700 
692019 
692338 
692656 
692976 
693293 
693612 
693930 
694248 
694566 
694883 
695201 
695518 
695836 
696153 
696470 
696787 
697103 
697420 
697736 
698053 


698685 
699001 
699316 
699632 
699947 
700263 
700678 
700893 

.701208 
701623 
701837 
702162 
702466 
702780 
703096 
703409 
703723 
704036 

. 704350 
704663 
704977 
705290 
705603 
705916 
706228 
706541 
706854 
707166 


_10^ 


53.4 

53.4 

53.4 

53.3 

53.3 

53.3 

63 

63 

63 

63 

63 

53 

53 

53.1 

53.1 

53.1 

53.0 

53.0 

53.0 

63.0 

62.9 

52.9 

62.9 

52.9 

52.9 

52.8 

52.8 

52.8 

62.8 

52.7 

52.7 

52.7 

52.7 

52.6 

52.6 

62.(5 

52.6 

52.6 

52.6 

62.5 

62.6 

62.4 

62.4 

52.4 

52.4 

52.4 

52.3 

52.3 

52.3 

52.3 

52.2 

52.2 

52.2 

52.2 

52.2 

52.1 

52.1 

62.1 

52.1 

52.1 


Cclang 
Degrees. 


Cotong.  j|N.  sine 


10.311818 
311498 
311177 
310857 
310537 
310217 
309897 
309677 
309268 
308938 
308619 

10.308300 
307981 
307662 
307344 
307026 
306707 
306388 
306070 
305752 
305434 

10.305117 
304799 
?J  04482 
304164 
303847 
303580 
303213 
302897 
302680 
302264 

10-301947 
301631 
301315 
300999 
300684 
300368 
300058 
299737 
299422 
299107 

10-398792 
298477 
298163 
297848 
297684 
297220 
296906 
2S6691 
296277 
295964 

10.2^5660 
295337 
295028 
294710 
294397 
294084 
293772 
293469 
29ol46 
292884 


Tang 


43837 
43863 
43889 
43916 
43942 
43968 
43994 
44020  89790 


N^oos 

89879 
89867 
89854 
89841 
89828 
89816 
89803 


44046 
44072 
44098 
44124 
44151 
44177 
44208 
44229 
44265 
44281 
44307 
44333 
44359 
44885 
44411 
44437 
44464 
44490 
44516 
44642 
44668 
44694 
44620 
44646 
44672 
44698 
44724 
44750 
44776 
44802 
44828 
44854 
44880 
449U6 
44932 
44958 
44984 
45010 
45036 
145062 
45088 
46114 
45140 
45166 
45192 
45218 
45248 
45269 
45295 
45321 
45347 
45373 
45599 
N .  «-os.  N.sine. 


89777 
89764 
89762 
89739 
89726 
89718 
89700 
89687 
89674 
89662 
89649 
89636 
89623 
89610 
89597 
89534 
89571 
89558 
89545 
89532 
89519 
89506 
89493 
89480 
89467 
89454 
89441 
89428 
89415 
89402 
89889 
89376 
89363 
89860 
89337 
89324 
89811 
89298 
89285 
89272 
89259 
89246 
89232 
89219 
89206 
b9193 
89180 
89167 
89153 
89140 
89127 
89114 
89101 


48 


Log.  Sines  and  Tangent*.    (27^)    Natural  Sines. 


TABLE  n. 


19.657047 
667295 
657642 
667790 
658037 
668284 
668531 
668778 
659025 
659271 
659517 

9.659763 
660009 
660256 
660601 
660746 
660991 
661236 
661481 
661726 
661970 

9.662214 
662469 
662703 
662946 
663190 
663433 
663677 
663920 
664163 
664406 

9.664648 
664891 
665133 
665375 
665617 
666859 
666100 
666342 
666583 
666824 

9.667065 
667305 
667546 
667786 
668027 
668267 
668506 
668746 
668986 
669225 
669464 
669703 
669942 
670181 
670419 
670668 
670896 
671134 
671372 
671609 


Cosine. 


D.  10 


41.3 
41.3 
41.2 
41.2 
41.2 
41.2 
41.1 
41.1 
41.1 
41.0 
41.0 
41.0 
40.9 
40.9 
40.9 
40.9 
40.8 
40.8 
40.8 
40.7 
40.7 
40.7 
40.7 
40.6 
40.6 
40.6 
40.5 
40.6 
40.5 
40.5 
40.4 
40.4 
40.4 
40.3 
40.3 
40.3 
40.2 
40.2 
40.2 
40.2 
40.1 
40.1 
40.1 
40.1 
40.0 
40  0 
40.0 
39.9 
39.9 
39.9 
39.9 
39.8 
39.8 
39.8 
39.7 
39.7 
39.7 
39.7 
39.6 
39.6 


.949881 
949816 
949762 
949688 
949623 
949558 
949494 
949429 
949364 
949300 
949235 

.949170 
949105 
949040 
948975 
948910 
948846 
948780 
948716 
948650 
948584 

.948619 
948454 
948388 
948323 
948267 
948192 
948126 
948060 
947995 
947929 

.947863 
947797 
947731 
947666 
947600 
947633 
947467 
947401 
947336 
947269 

.947203 
947136 
947070 
947004 
946937 
946871 
946804 
946738 
946671 
946604 

.946538 
946471 
946404 
946337 
946270 
946203 
946136 
946069 
946002 
945935 


Sine. 


). 707166  _» 

707478  IJ.' 

707790  IJ.' 

708102  It' 

708414  °f 

708726  °j 

709037  ?{• 

709349  °J 

709660  l\ 

709971  l\- 

710282  °J" 

>. 710593  l\- 

710904  l\' 

711216  °;- 

711525  °;- 
711836  l\- 
712146  °J- 
712456  l\- 
712766  l\- 
713076  ^}- 
713386  l\- 

>. 713696  l\- 
714006  l\- 
714314  l\- 
714624  l\- 
714933  °f- 
.715242  l\- 
715651  °J- 
716860  l\- 
716168  l\- 
716477  °} • 

1.716786  °)- 
717093  I  j;  • 
717401 1 1\ ■ 
717709  l\- 
718017  l\- 
718325  ^f- 
718633  l\- 
718940  l\- 
719248  ?;• 
719555  ^|- 

.719862  ^;- 
720169  °}- 
720476  °}- 
720783  l\- 
721089  ?}• 
721396  °\- 
721702  v.- 
722009  2|- 
722316  °}- 
722621  ?J- 

1.722927  °\- 

723232  °/;- 

723638 ;°"- 
723844^^. 

7241491^0- 
724454  I °" • 
724759  I °" • 
725066;°"- 
726369^2- 
726674  p"- 


Co  tang. 
Degreea. 


Cotang.  I  N.  sine.  N.  cos, 


10 


10 


10 


10 


.292834 
292622 
292210 
291898 
291686 
291274 
290963 
290651 
290340 
290029 
289718 
.289407 
289096 
288785 
288475 
288164 
287864 
287644 
287234 
286924 
286614 
.286304 
285996 
286686 
285376 
286067 
284758 
284449 
284140 
283832 
283523 
,283215 
282907 
282599 
282291 
281983 
281675 
281367 
281060 1 
280752 1 
280445  i 
280138 
279831 1 
279524 , 
279217 
278911 
278604 
278298 
277991 
277686 
277379 
277073 
276768 
276462 
276156 
275861 
275546 
276241 
274935 
274631 
274326 


Tiing. 


45399 
46426 
45451 
45477 
45503 
45529 
46664 
45680 
45606 
45632 
45668 
45684 
45710 
46736 
45762 
45787 
45813 
45839 
45865 
45891 
45917 
45942 
45968 
45994 
46020 
46046 
46072 
46097 
46123 
46149 
46175 
46201 
46226 
46262 
46278 
46304 
46330 
46356 
46381 
46407 
4<)433 
46458 
46484 
46510 
46536 
46561 
46587 
46613 
46639 
46664 
46690 
46716 
46742 
46767 
46793 
46819 
46844 
46870 
46896 
46921 
46947 


N.  COF.  .^.^iln 


89101 
89087 
89074 
89061 
89048 
89036 
89021 
89008 
88995 
88981 
88968 
88955 
88942 
88928 
88916 
88902 
88888 
88875 
88862 
88848 
88835 
88822 
88808 
88795 
88782 
88768 
88766 
88741 
88728 
88715 
88701 
88688 
88674 
88661 
88647 
88634 
88620 
88607 
88593 
88580 
88566 
88653 
88639 
88526 
88512 
88499 
88485 
88472 
88458 
88445 
88431 
88417 
88404 
88390 
88377 
88363 
88849 
88336 
88322 
88308 
88295 


TABLE  II. 


Log.  Sines  and  Tangents.    (28°)    Natural  Smes. 


37 


60 


Sine. 

9.671609 
671847 
672034 
672321 
672558 
672795 
673032 
673268 
673505 
673741 
673977 

9.674213 
674448 
674684 
674919 
675155 
675390 
675624 
675859 
676094 
676328 

9.676562 
676796 
677030 
677264 
677498 
677731 
677964 
678197 
678430 
6786S3 

9.678895 
679128 
679360 
679592 
679824 
680056 
680288 
680519 
680750 
680982 

9.681213 
681443 
681674 
681905 
682135 
682365 
682595 
682825 
683055 
683284 
.683514 
683743 
683972 
684201 
684430 
684658 
684887 
685115 
685343 
685571 


D.  10" 


Cosino. 


Cosine. 


1.945935 
945868 
945800 
945733 
945666 
945598 
945531 
945464 
945396 
945328 
945261 
.945193 
945125 
945058 
944990 
944922 
944854 
944786 
944718 
944650 
944582 
.944514 
944446 
944377 
944309 
944241 
944172 
944104 
944036 
943967 
943899 
.943830 
943761 
943693 
943624 
943556 
943486 
943417 
943348 
943279 
943210 
.943141 
9-13072 
943003 
942934 
942864 
942795 
942726 
942656 
942587 
942517 
.942448 
942378 
942308 
942239 
942169 
942099 
942029 
941959 
941889 
941819 


D.  10' 


Sine. 


Tang. 

.725674 
725979 
726284 
726588 
726892 
727197 
727501 
727805 
728109 
728412 
728716 
.729020 
729323 
729626 
729929 
730233 
730535 
730838 
731141 
731444 
731746 
.732048 
732351 
782653 
732955 
733257 
733558 
733860 
734162 
734463 
734764 
.735066 
735367 
735668 
785969 
736269 
736570 
736871 
737171 
737471 
737771 
738071 
738371 
738671 
738971 
789271 
739570 
739870 
740169 
740468 
740767 
741066 
741365 
741664 
741962 
742261 
742559 
742858 
743156 
743454 
743752 


D,  10' 


Cotang.  I  N.  sine.lN.  cos 


Cotang. 


47690 
47716 
47741 


1 47767  87854 
47793  87840 
4781837826 


47844 
47869 


47920 
47946 
47971 
47997 


87896 
87882 
87868 


87812 
87798 


47895  87784 


87770 
87756 
87743 
87729 


48022(87715 
48048  87701 
i  48073  87687 
48099  87673 


48124 


87659 


4815087645 
48175  87631 


48201 


48303 
48328 
48354 


48430 
48456 
48481 


Tang.   I  N.  coo.  N.sine 


87617 


48226  87603 
48252|87589 
48277  87575 


87561 
87546 
87532 


48379  87518 
48405  87504 


87490 
87476 
87462 


37 


61  Degrees. 


50 


Log.  Sines  and  Tangents.    (29°)    Natural  Sines. 


TABLE  n. 


0 
1 

2 
3 
4 
6 
6 
7 
8 
9 
10 
11 
12 
13 
14 
16 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
i29 
30 
31 
32 
33 
34 
36 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
48 
47 
48 
49 
60 
51 
52 
63 
54 
56 
66 
57 
58 
69 
60 


D.  W 


685571 
686799 
686027 
686254 
686482 
686709' 
686936 
687163 
687389 
687616 
687843 
9.688069 
688295 
688521 
688747 
688972 
689198 
689423 
689648 
689873 
690098 
9.690323 
690548 
690772 
690996 
691220 
691444 
691668 
691892 
692116 
692339 
9.692562 
692785 
693008 
693231. 
693463 
693676 
69389& 
694120 
694342 
694564 
9.694786 
695007 
696229 
695460 
696671 
695892 
696113 
696334 
696664 
696775 
9.6969y5 
697215 
697435 
697654 
697874 
698094 
698313 
698532 
698751 
698970 
Coaine. 


38.0 

37.9- 

37.9 

37.9 

37.9 

37.8 

37.8 

37.8 

37.8 

37.7 

37.7 

37.7 

37.7 

37.6 

37.6 

37.6 

37.6 

37.6 

37.5 

37,5 

37.5 

37.4 

37.4 

37.4 

37.4 

37.3 

37.3 

37,3 

37,3 

37.2 

37.2 

37.2 

37.1 

37.1 

37.1 

37.1 

37.0 

37,0 

37.0 

37.0 

36.9 

36.9 

36.9 

36,9 

36.8 

36.8 

36.8 

36.8 

36.7 

36.7 

36.7 

36.7 

36.6 

36.6 

36.6 

36.6 

36.6 

36.6 

36.6 

36,6 


Cosine.  \D.  IC/' 


9.941819 
941749 
941679 
941609 
941639 
941469 
941398 
941328 
941258 
941187 
941117 
941046 
940976 
940905 
940834 
940763 
940693 
940622 
940651 
940480 
940409 
9.940338 
940267 
940196 
940126 
940054 
939982 
939911 
939840* 
939768 
939697 
939625 
939654 
939482 
939410 
939339 
939267 
939195 
939123 
939052 
938980 
9.938908 
938836 
938763 
938691 
938619 
938647 
938476 
938402 
938330 
938268 
9G8185 
938113 
938040 
937967 
937895 
937822 
937749 
937676 
937604 
937531 


'11. 7 
11.7 
11.7 
11.7 
11.7 


Sine. 


11 

11 

11 

11 

11 

11 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 


11.8 
11.8 
11.3 
11.8 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.-9 
12.0 
12.0 
12.0 
12.0 
12.0- 
12.0 
12.0- 
12.0 
12.0 
12.0 
12.0 
12.0 
12.1 
12.1 
12,1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 


Tang.  D.  10" 


9,743752 
744050 
744348 
744646 
744943 
746240 
745638 
745836' 
746132 
746429 
746726 

9.747023 
747319 
747616 
747913 
748209 
748505 
748801 
749097 
749393 
749689 

9.749985 
75(0281 
760676 
760872 
751167 
761462 
751767 
752062 
762347 
752642 

9.752937 
753231 
753626 
753820 
754116 
764409 
754703 
754997 
756291 
765685 

9,765878 
756172 
766465 
766759 
757062 
767346 
767638 
757931 
768224 
768617 

9',  768810 
769102 
759395 
759687 
759979 
760272 
760564 
760866 
761148 
761431 
Cotang. 


49.6 
49.6 
49.6 
49.6 
49.6 
49.5 
49.5 
49,5 
49,5 
49.5 
49.4 
49.4 
49.4 
49.4 
49^4 
49.3 
49.3 
49.3 
49,3 
49.3 
49.3 
4Q,2 
49.2 
49.2 
49.2 
49.2 
49.2 
49.1 
49.1 
49,1 
49,1 
49.1 
49.1 
49.0 
49.0 
49.0 
49.0 
49.0 
49.0 
48.9 
48,9 
48,9 
48.9 
48.9 
48.9 
48.8 
48  ..8 
48.8 
48.8 
48,8 
48.8 
48.7 
48.7 
48.7 
48,7 
48,7 
48.7 
48.6 
48.6 


Cotang. 


10.256248 
266950 
255652 
256356 
266067 
254760 
264462 
254166 
253868 
263571 
263274 

10.252977 
252681 
252384 
252087 
251791 
261496 
251199 
250903 
260607 
250311 

10.260015 
249719 
249424 
249128 
248833 
248638 
248243 
247948 
247653 
247358 

10.247063 
246769 
246474 
246180 
245886 
246691 
245297 
246003 


N,8ine, 


48481 
48606 
48532 
48667 
48583 
48608 
48634 
48659 
48684 


87462 
87448 
87434 
87420 
87406 
87391 
87377 
87363 
87349 


4871087335 


48736 
48761 
48786 
48811 
48837 
48862 
48888 
48913 


48964 
48989 
49014 


49066 
49090 
49116 
49141 
49166 
49192 
49217 
49242 
49268 
49293 
49318 
49344 
49369 


87321 
87306 
87292 
87278 
87264 
87260 
87235 
87221 
87207 
87193 
87178 
87164 


49040187150 


49419 
49446 


87136 
87121 
87107 
87093 
87079 
87064 
87050 
87036 
87021 
87007 
86993 
86978 
86964 
86949 
86935 
86921 


244709  4947086906 


244416  11 49495 
10.2441221!  49621 
243828  149646 
24363&i|49571 
243241  49596 


342948 
242655 
242362 
242069 
241776 
241483 

10.241190 
240898 
240606 
240313 
240021 
239728 
239436 
239144 
238852 
238661 

I   Tang. 


1 49622 
49647 
49672 
49697 
49723 
49748 
49773 
49798 


86892 
86878 
86863 
86849 
86834 
86820 
86805 
86791 
86777 
86762 
86748 
86733 
86719 


4982486704 
49849j86690 
4987486675 


49899 
49924 


86661 
86646 


4995086632 


49976 
50000 


N.  cofi.  N.Hiiic 


86617 
86603 


60 
69 
58 
57 
66 
55 
54 
63 
52 
61 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


60  Degreeg. 


TABLE  n. 


Log.  Sines  and  Taugeats.    (30°)    Natural  Sines. 


51 


Sine. 


D.  10" 


9.698970 
699189 
699407 
699626 
699844 
700062 
700280 
700498 
700716 
700933 
701151 j 

9.701368 
701585 
701802 
702019 
702236 
702462 
702669 
702886 
703101 
703317 
703533 
703749 
703964 
704179 
704395 
T04610 
704825 
705040 
705254 
705469 

9.706683 
705898 
706112 
706326 
706539 
706753 
706967 
707180 
707393 
707606 
.707819 
708032 
708245 
708458 
70S670 
708882 
709094 
709308 
709518 
709730 
.709941 
710163 
710J64 
710576 
710786 
710967 
711208 
711419 
711629 
711839 
Cosine. 


36.4 
36.4 
36.4 
36.4 
36.3 
36.3 
36.3 
36.3 
36.3 
36.2 
36.2 
36.2 
36.2 
,1 


36 

36.1 

36.1 

36.1 

36.0 

36.0 

36.0 

36.0 

36.9 

35.9 

36.9 

36.9 

36.9 

36.8 

35.8 

36.8 

35.8 

35.7 

35.7 

35.7 

35.7 

36.6 

35.6 

35.6 

35.6 

35.5 

36.6 

35.6 

35.6 

35.4 

35.4 

35.4 

35.4 

35.3 

36.3 

35.3 

35.3 

36.3 

36.2 

35.2 

35.2 

36.2 

35.1 

35.1 

35.1 

35.1 

35.0 


Cosine. 


D.  10'' 


12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12,3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.5 

12.6 

12,6 

12,6 

12.5 

12.6 

12.5 


12 

12 

12 

12 

12 

12 

12 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 


Tanp" 


.761439 
761731 
762023 
762314 
762G06 
762897 
763188 
763479 
763770 
764061 
764352 

.764643 
764933 
766224 
765614 
766806 
766095 
766385 
766676 
766965 
767265 

.767546 
767834 
768124 
768413 
768703 
768992 
769281 
769570 
769860 
770148 

.770437 
770726 
771015 
771303 
771692 
771880 
772168 
772457 
772745 
773033 

.773321 
773608 
773896 
774184 
774471 
774759 
775046 
775333 
775621 
776908 

.776195 
776482 
776769 
777065 
777342 
777628 
777915 
778201 
778487 
778774 


10' 


48.6 


48.6 


Cotang. 


Cotang.     j  N.  sine.  N.  cos 


10.238561 
238269 
237977 
237686 
237394 
237103 
236812 
236521 
236230 
235939 
235648 

10.235357 
235037 
234776 
234486 
234195 
233905 
233615 
233325 
233035 
232746 

10.232455 
232166 
231876 
231587 
231297 
231008 
230719 
230430 
230140 
229852 

10.229563 
229274 
228985 
228697 
228408 
228120 
227832 
227543 
227256 
22696.7 

10.226679 
226392 
226104 
225816 
226529 
225241 
224954 
224667 
224379 
224092 

10.223806 
223518 
223231 
222945 
222658 
222372 
222085 
221799 
221612 
221226 


150000 
50025 
60050 
50076 
50101 
60126 
60151 
50176 
60201 
50227 
50252 
50277 
50302 
50327 
50352 
50377 
50403 
1 50428 
'60453 
60478 
50603 
60528 


86588 
86573 
86559 
86544 
86630 
86616 
86601 
86486 
86471 
86457 
86442 
86427 
86413 
86398 
86384 


Tang. 


86354 
86340 
86325 
86310 
86295 


60653  86281 


60678 

50603 

50628 

50654 

50679 

50704 

60729 

50754 

60779 

60804 

60829 

.50864 

.5087: 

50904 

50929 

50954 

50979 

51004 

51029 

51054 

51079 

61104 

51129 

61154 

51179 

51204 

51229 

5126<i 

61279 

5J304 

51329 

51364 

51379 

51404 

51429 

51454 

61479 

61504 


86266 
86251 
86237 
86222 
86207 
86192 
86178 
86163 
86148 
86133 
86119 
86104 
986089 
86074 
86059 
86045 
86030 
86015 
86000 
85985 
85970 
85956 
85941 
86926 
85911 
85896 
85881 
85866 
85861 
85836 
85821 
85806 
85792 
85777 
86762 
86747 
85732 
85717 


N.  COS.  N.8ine 


59  Degrees. 


52 


Log.  Sine8  and  Taagenta.  (31°)  Xatural  Sines. 


TABLE  II. 


Sine.  |D.  10"  (Josine, 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 

la 
11 

12 

13 

14 

15 

16 

17 

18 

19 

20  i 

21 

22 

23 

24 

25 

26 

27 

2S 

29 

30  i 

31  9 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41  9 
42 
43 
44 
45 
46 
47 
48 
49 
50 
61 
62 
53 
54 
55 
56 
57 
68 
59 
60 


.711839  OK 

712050  ^? 

712260  r.~ 

712469  ^° 

712679  tri 

712889  ^] 

713098  ■^l 

713308  t>l 

713617  t: 

713726  ^l 

713936  tl 

.714144  tj 

714352  ll 

714561  tl 

714769  t; 

714978  ^2 

715186  '^2 

715394  ^^ 

7166Q2  ^2 

715809  "il 

716017  ^* 

.716224  fl 

716432  q^ 

716639  ^2 

716846  r: 

717053  ^^ 

717259  tl 

717466  ^t 

717673  ^;: 

717879  t: 
718086  ^^ 

.718291  "ij 
718497  ^J 
718703  ^^ 
718909  i  '11 
719114  ^^ 
719320  7.1 
719525  ^;: 
719730  f: 
719936  ^^ 
720140  ^? 

.720346  ll 
720549 !^1' 
720754;^^' 
720958  I  tl 
721162' 5; 
721366 1  tl 
721570  ^X 
721774 ■ it 
721978:^ 
722181'^^ 

.  722385 ^^t 
722588  ^^ 
722791!^^ 
722994^^ 
723197:^^ 
723400  „ 
723603  to 
723805^^ 
724007^^ 
724210!  ^ 
Cosine.  I 


9.933036 
932990 
932914 
932838 
932762 
932685 
932609 
932533 
932457 
932380 
932304 

9.932228 
932151 
932076 
931998 
931921 
931845 
931768 
931691 
931614 
931537 
931460 
931383 
931306 
931229 
931152 
931075 
930998 
930921 
930843 
930766 

9.930688 
930611 
930533 
930466 
930378 
930300 
930223 
930145 
930067 
929989 
929911 
929833 
929766 
929677 
929599 
929521 
929442 
929364 
929286 
929207 
929129 
929050 
928972 


928816 
928736 
928657 
928678 
928499 
928420 
Sine. 


D.  10" 


12.6 
12.7 

12.7 


12 

12 

12 

12 

12 

12 

12.7 

12.7 

12.7 

12.7 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.8 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

12.9 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.0 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 

13.1 


Tang. 


1.778774 
779060 
779346 
779632 
779918 
780203 
780489 
780775 
781060 
781346 
781631 

1.781916 
782201 
782486 
782771 
783053 
783341 
783626 
783910 
784195 
784479 

(.784764 
785048 
785332 
785616 
785900 
786184 
786468 
786752 
787036 
787319 

>.  787603 
787886 
788170- 
788453 
788736 
789019 
789302 
789685 
789868 
790151 

1.790433 
790716 
790999 
791281 
791663 
791846 
792128 
792410 
792692 
792974 

). 793256 
793638 
793819 
794101 
794383 
794664 
794946 
795227 
795508 
796789 
Cotang. 


D.  ll>"  Cotang.   N.sine.  N.  cos 


47.7 

47.7 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.6 

47.5 

47.5 

47.4 

47 

47 

47 

47 

47 

47 

47.3 

47.3 

47.3 

47.3 

47.3 

47.3 

47.2 

47.2 

47.2 

47.2 

47.2 

47.2 

47.2 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.1 

47.0 

47.0 

47.0 

47.0 

47.0 

47.0 

47.0 

46.9 

46.9 

46.9 

46.9 

46.9 

46.9 

46.9 

46.8 


10.221226 
220940 
220664 
220368 
220082 
219797 
2195U 
219226 
218940 
218654 
218369 

10.218084 
217799 
217514 
217229 
216944 
216669 
216374 
216090 
215805 
215521 

10.215236 
214952 
214668 
214384 
214100 
213816 
213532 
213248 
212964 
212681 

10.212397 
212114 
211830 
211647 
211264 
210981 
210698 
210416- 
210132 
209849 

10.209567 
209284 
209001 
208719 
208437 
208164 
207872 
207690 
207308 
207026 

10.206744 
206462 
206181 
205899 
205617 
205336 
205055 
204773 
204492 
204211 


51504  86717 
61529  86702 
61554  85687 
51579  85672 


61604 
51628 
61653 
51678 
! 61703 
161728 
1  61753 
'I  61778 
61803 
51828 
61852 
61877 
51902 
51927 
61952 


85667 
85642 
85627 
86612 
86697 
85682 
85567 
85551 
86536 
85521 
85606 
86491 
86476 
86461 
86446 


61977185431 

62002185416 

52026J85401 

52061 

62076 


86386 

85370 

52101^85365 


6Q126 
62161 
152175 
! 52200 
! 62226 
i  52260 
i  52276 


85340  j  36 
85325  I  34 
86310  33 


85294 
85279 
85264 
86249 
52299  85234  28 
62324J85218  27 
i  52349  85203  26 


i  52374 
62399 
; 62423 
152448 
i  52473 
I  62498 


85188  1 26 


86173 
86167 
85142 
85127 
86112 


I  52522  85096 

i  52547  '^  ■"' 
1 62572 


! 62597 
1 62621 
i  52646 
■62671 
! 52696 
i  62720 
i  62746 
I  52770 
i 52794 
162819 
62844 


52869  8488; 
52893  84866 
52918  84861 


85U81 
85066 
85U51 
86036 
85020 
85005 
84989 
84974 
84959 
84943 
849:^8 
84913 
84897 


!  62943 
j!  62967 
1162992 


Tang. 


84836 

84820 

84805 

N.  cos.JN.sine. 


58  Degrees. 


TABLE  n. 


Log.  Sines  and  Tangents.    (32°)    Natural  SincB. 


53 


Sine. 


9.724210 
724412 
724614 
724816 
725017 
725219 
725420 
725622 
725823 
726024 
726225 
726426 
726626 
726827 
727027 
727228 
727428 
727628 
727828 
728027 
728227 

9.728427 
728626 
728825 
729024 
729223 
729422 
729621 
729820 
730018 
730216 

9.730415 
730613 
730811 
731009 
731206  I 
731404 
731602 
731799 
731996 
732193 

9.732390 
732587 
732784 
732980 
733177 
733373 
733569 
733765 
733961 
734157 

9.734353 
734549 
734744 
734939 
735135 
735330 
735525 
735719 
735914 
736109 


D.  10" 


Ck)sine. 


Cosine.  |D.  ly^ 


9.928420 
928342 
928263 
928183 
928104 
928025 
927946 
927867 
927787 
927708 
927629 
927549 
927470 
927390 
927310 
927231 
927151 
927071 
926991 
92691 1 
926831. 

9.926751 
926671 
926591 
926511 
926431 
926351 
926270 
926190 
926110 
926029 
926949 
925868 
926788 
925707 
925626 
925545 
925465 
925384 
926303 
925222 
925141 
925060 
924979 
924897 
924816 
924735 
924654 
924572 
924491 
924409 

9.924328 
924246 
924164 
924083 
924001 
923919 
923837 
923755 
923673 
923591 


Sine. 


13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13.3 

13.3 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 


13.4 


13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.6 

13.7 

13.7 


Tang. 


). 795789 
796070 
796351 
796632 
796913 
797194 
797475 
797765 
798036 
798316 
798596 

).  798877 
799157 
799437 
799717 
799997 
800277 
800557 
800836 
801116 
801396 

). 801676 
801956 
802234 
802613 
802792 
803072 
803361 
803630 
803908 
804187 

). 804466 
804745 
805023 
806302 
805680 
806859 
806137 
806415 
806693 
806971 

).  807249 
807527 
807805 
80S083 
808361 
808638 
808916 
809193 
809471 
809748 

).  810025 
810302 
810580 
810857 
811134 
811410 
811687 
811964 
812241 
812517 

Cotanc. 


D.  10" 


46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.7 
46.7 
46.7 
46.7 
46.7 
46.7 
46.7 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.5 
46.5 
46.5 
46.5 
46.5 
46.5 
46.5 
46.5 
46.4 
46.4 
46.4 
46.4 
46.4 
46.4 
46.4 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.1 
46.1 
46.1 
46.1 
46.1 


Cotang.  I  N.  sine.  N.  cos 


10.204211 
203930 
203649 
203368 
203087 
202806 
202525 
202245 
201964 
201684 
201404 

10.201123 
200843 
200563 
200283 
200003 
199723 
199443 
199164 
198884 
198604 

10.198326 
198046 
197766 
197487 
197208 
196928 
196649 
196370 
196092 
195813 

10.195534 
195255 
194977 
194698 
194420 
194141 
193863 
193686 
193307 
193029 

10.192751 
1S2473 
192195 
191917 
191639 
191362 
191084 
190807 
190529 
190252 
189975 
189698 
189420 
189143 
188806 
188590 
188313 
188036 
187759 
187483 

"Tani?. 


52992 
53017 
53041 
53066 
53091 
53115 
53140 
153164 
53189 
53214 
53238 
53263 


53312 
53337 
53361 
63386 
53411 
53435 
1 53460 
1 53484 
63509 
1 63634 
1 53568 
i  63683 
1 53607 
53632 
53656 
53681 
53706 
63730 
63754 
53779 


10. 


84805 
84789 
84774 
84759 
84743 
84728 
84712 
84697 
84681 
84666 
84650 
84635 
84619 
84604 
84588 
84573 
84657 
84542 
84626 
84511 
84495 
84480 
84464 
84448 
84433 
84417 
84402 
84386 
84370 
84355 
84339 
84324 
84308 
53804  84292 
1 53828J84277 
i  63853184261 
1 53877[84245 
1 53902  84230 
!  63926  84214 
!  63951  84198 
I53975s84l82 
54000  84167 
I  64024|84151 
154049  84135 
1 54073,84120 
54097184104 
1 54122J84088 
i  54146184072 
1 54171  84057 
i  64195  84041 
1 54220,84025 
I  64244:84009 
154269183994 
54293  83978 
1 54317  83962 
54342,8S946 
!  54366  83930 
1 54G91 183915 
!  64415183899 
;  64440;83883 
15446483867 


60 
59 
58 
57 
66 
55 
54 
53 
62 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
S3 
32 
31 
30 
29 
28 
27 
26 
26 
24 
23 
^2 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
S 
2 
1 
0 


N.  COP.  N.fiir.e. 


57  Degrees. 


54 


Log.  Sines  and  Tangents.  (33°)  Natural  Sines.     TABLE  IL 


J_ Sine.   D.  V^'     Cosine.  D.  W      Tang.   D.  10"  Cotang 


8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

64 

55 

56 

57 

58 

59 

60 


9.736109 
736303 
736498 
736692 
736886 
737080 
737274 
737467 
737661 
737855 
738048 
9.738241 
738434 
738627 
738820 
739013 
739206 
739398 
739590 
739783 
739975 
740167 
740369 
740550 
740742 
740934 
741125 
741316 
741508 
741699 
741889 
9.742080 
742271 
742462 
742662 
742842 
743033 
743223 
743413 
743602 
743792 
19.743982 
744171 
744361 
744560 
744739 
744928 
745117 
745306 
745494 
745683 
9.745871 
746059 
746248 
746436 
746624 
746812 
746999 
747187 
747374 
747562 
Cosine. 


32.4 

32.4 

32.4 

32.3 

32.3 

32.3 

32.3 

32.3 

32.2 

32.2 

82.2 

32.2 

32.2 

32.1 

32.1 

32.1 

32.1 

32.1 

32.0 

32.0 

32.0 

32.0 

32.0 

31.9 

31.9 

31.9 

31.9 

31.9 

31.8 

31.8 

31.8 

31.8 

31 


31.7 
31.7 
31.7 
31.7 
31.7 
31.6 
31.6 
31.6 
31.6 
31.6 


31.5 
31.5 
31.5 
31.5 
31.5 
31.4 
31.4 
31.4 
31.4 
31.4 
31.3 
31.3 
31.3 
31.3 
31.3 
31.2 


31.2 


9.923591 
923509 
923427 
923345 
923263 
923181 
923098 
923016 
922933 
922851 
922768 
9.922686 
922603 
922520 
922438 
922355 
922272 
922189 
922106 
922023 
921940 
9.921857 
921774 
921691 
921607 
921624 
921441 
921357 
921274 
921190 
921107 
9.921023 
920939 
920856 
920772 
920688 
920604 
920520 
920436 
920362 
920268 
9.920184 
920099 
920015 
919931 
919846 
919762 
919677 
919593 
919508 
919424 
9.919339 
919254 
919169 
919086 
919000 
918915 
918830 
918745 
918659 
918574 


I   Sine. 


13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.2 

14.2 

14.2 

14.2 


9.812517 
812794 
813070 
813347 
813623 


814175 
814452 
814728 
815004 
815279 
9.816555 
815831 
816107 
816382 
816658 
816933 
817209 
817484 
817759 
818036 
.818310 
818685 
818860 
819135 
819410 
819684 
819959 
820234 
820608 
820783 
9.821067 
821332 
821606 
821880 
822154 
822429 
822703 
822977 
823250 
823524 
3.823798 
824072 
824345 
824619 
824893 
825166 
825439 
826713 
825986 
826259 
).  826632 
826805 
827078 
827351 
827624 
827897 
828170 
828442 
828715 
828987 
Cotang. 


46.1 

46.1 

46.1 

46.0 

46.0 

46.0 

46.0 

46 

46 

46 

46.0 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.9 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.8 

45.7 

45.7 

45.7 

45.7 

46.7 

45.7 

45.7 

45.7 

45.7 

45.6 

46.6 

45.6 

45.6 

46.6 

46.6 

45.6 

45.6 

45.6 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.5 

45.4 

45.4 

45.4 

45.4 


10.187482 
187206 
186930 
186653 
186377 
186101 
185825 
185548 
185272 
184996 
184721 
10.184445 
184169 
183893 
183618 
183342 
183067 
182791 
182516 
182241 
181965 
10.181690 
181415 
181140 
180865 
180590 
180316 
180041 
179766 
179492 
179217 
10.178943 
178668 
178394 
178120 
177846 
177571 
177297 
177023 
176760 
176476 
10.176202 
175928 
176656 
175381 
175107 
174834 
174561 
174287 
174014 
173741 
10.173468 
173195 
172922 
172649 
172376 
172103 
171830 
171558 
171285 
171013^ 
Tai^. 


N.  sine.  N.  cos. 


54464 
54488 
54513 
54537 
54661 


83867 
83851 
83835 
83819 
83804 


54586  83788 


54610 


83772 


54635  83756 
54659  83740 
54683  «3724 


54708 
64732 
54766 
54781 


54805  83645 
5482983629 
54854  83613 


54878 
54902 
54927 


83697 
83581 
83566 


54951  83549 


54975 
54999 
55024 


56097 
55121 
55145 
55169 
65194 
55218 
65242 


83708 
83692 
83676 
83660 


83533 
83517 
83601 


56048  83485 
55072  83469 


83463 
83437 
83421 
83406 
83389 
83373 
83356 


5626683340 


56291 
65315 
55339 


56412 
65436 
56460 
56484 
55509 
56533 
55657 
56581 


83324 


83292 


65363  83276 
83260 
83244 
83228 
83212 
83195 
83179 
83163 
83147 
83131 
56605  83116 
55630  ,'83098 
55654  83082 
56678|83066 
65702J83060 
55726  83034 
5675083017 


55871 
55895 
55919 


83001 
82S85 


55775 
66799 
55823 
55847  82953 


82936 
82920 
82904 


N.  eos.  N.sine, 


60 
59 
68 
67 
56 
65 
64 
53 
52 
61 
60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
16 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 


56  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (34°)    Natural  Sinea- 


55 


I).  10" 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
36 
36 
37 
38 
39 
40 
41 
42 
43 
44 
46 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 

9.747562 
747749 
747936 
748123 
748310 
748497 
748683 
748870 
749056 
749243 
749426 

9.749615 
749801 
749987 
750172 
750358 
760543 
750729 
750914 
761099 
751284 

9.751469 
761654 
751839 
762023 
752208 
752392 
752576 
752760 
752944 
753128 

9.763312 
753495 
763679 
753862 
764046 
764229 
754412 
754595  I 
764778  '■ 
754960  I 

9.7661431 
766326  I 
755608  I 
765690 : 
755872  j 
756054  i 
756236  I 
756418 ! 
756600  i 
756782 ! 

9.756963! 
757144  I 
767326 
757507  I 
757688  I 
767869 : 
758050 : 
758230  i 
768411 ! 
_758591 ! 
Cosine.  I 


31.2 
31.2 
31.2 
31.1 
31.1 
31.1 
31.1 
31.1 
31.0 
31.0 
31.0 
31.0 
31.0 
30.9 
30.9 
30.9 
30.9 
30.9 
30.8 
30.8 
30.8 
30.8 
30.8 
30.8 
30.7 
30.7 
30.7 
30.7 
30.7 
30.6 
30.6 
30.6 
30.6 
30.6 
30.5 
30.5 
30.6 
30.5 
30.6 
30.4 
30.4 


30 

30 

30 

30 

30 

30 

30.3 

30.3 

30.3 

30.2 

30.2 

30.2 

30.2 

30.2 

30-1 

30.1 

30.1 

30.1 

30.1 


Cosine. 

9.918674 
918489 
918404 
918318 
918233 
918147 
918082 
917976 
917891 
917805 
917719 

9.917634 
917648 
917462 
917376 
917290 
917204 
917118 
917032 
916946 
916859 
.916773 
916687 
916600 
916514 
916427 
916341 
916254 
916167 
916081 
915994 
.916907 
915820 
916733 
915646 
915559 
915472 
915385 
915297 
915210 
915123 
.915036 
914948 
914860 
914773 
914686 
914698 
914610 
914422 
914334 
914246 
9.914158 
91407.0 
913982 
913894 
913806 
913718 
913630 
913541 
913453 
913365 
Sine. 


D.  10"   Tang. 


14.2 
14.2 
14.2 
14.2 
14.2 
14.2 
14.2 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 
14.3 


14 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14.4 

14.4 

14 

14 

14 

14 

14 

14 

14 

14 

14 

14.5 

14.5 

14.5 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.7 

14.7 

14. 

14. 

14. 

14. 

14. 

14. 

14. 


7 
7 
7 
7 
7 
7 
7 
14.7 


,828987 
829260 
829532 
829806 
830077 
830349 
830621 
830893 
831166 
831437 
831709 
831981 
832263 
832626 
832796 
833068 
833339 
833611 
833882 
834154 
834426 
834696 
834967 
836238 
835509 
835780 
836061 
836322 
836693 
836864 
837134 

9.837406 
837675 
837946 
838216 
838487 
838757 
839027 
839297 
839568 
839838 

9.840108 
840378 
840647 
840917 
841187 
841457 
841726 
841996 
842266 
842635 

9.842805 
843074 
843343 
843612 
843882 
844161 
844420 
844689 
844958 
846227 
Cotang. 


D.  10 


45.4 

45.4 

45.4 

45.4 

45.4 

45.3 

45.3 

46.3 

45.3 

45.3 

46.3 

45.3 

46.3 

45.3 

45.3 

46.2 

45.2 

45.2 

45.2 

45.2 

46.2 

45.2 

46.2 

46.2 

45.2 

45.1 

45.1 

46,] 

45.1 

45.1 

46.1 

46.1 

45.1 

45.1 

45.1 

46.0 

45.0 

45.0 

45.0 

45.0 

45.0 

46.0 

45.0 

45.0 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.8 

44.8 

44.8 

44.8 

44.8 


Cotang.    IjN.sine 


55919 
56943 
55908 
56992 
56016 
56040 
56064 
56088 
56112 
56136 
66160 


10. 16-8019!!  56184 


N.  COS.) 


167747 ! 

167475  i 

167204 

166932 

166661 

166389 

166118 

165846 

165576 

10.165304 
166033 
164762 
164491 
164220 
163949 
163678 
163407 
163136 
162866 

10.162595 
162325 
162054 
161784 
161513 
161243 
160973 


56208 
56232 
56256 
66280 
56305 
56329 
56353 
56377 
56401 
56425 
56449 
56473 
56497 
56621 
56545 
56569 
56593 
56617 
66641 
56665 
66689 
66713 
56736 
56760 
56784 
66808 


160703  !i568S2 
16043211 56856 
160162  1 156880 


10.159892 
169622 
159353 
159083 
158813 
168543 
158274 
168004 
157734 
157465 

10.157196 
166926 
166657 
156388 
156118 
155849 
155580 
155311 
155042 
154773 
Tang. 


56904 
56928 
56962 
56976 
57000 
67024 
57047 
67071 
57095 
57119 
57143 
67167 
67191 
57215 
57238 
57262 
67286 
67310 
57334 
57358 


82904 

82887 

82871 

82855 

82839 

82822 

82806 

82790 

82773 

82767 

82741 

82724 

82708 

82692 

82675 

82659 

82643 

82626 

82610 

82593 

82577 

82661 

82544 

82528 

82511 

82495 

82478 

82462 

82446 

82429 

82413 

82396 

82380 

82363 

82347 

82330 

82314 

82297 

82281 

82264 

82248 

82231 

82214 

82198 

82181 

82165 

82148 

82132 

82115 

82098 

82082 

82065 

82048 

82032 

82015 

81999 

81982 

81965 

81949 

81932 

81915 


N.  COS.  N.sine, 


56 


Log.  Sines  and  Tangents.    (35°)    Natural  Sines. 


TABLE  II, 


9.758591 
758772 
758952 
759132 
759312  "V 
759492  i:{"  • 
759672^"- 

759852  ;;;!• 
760031  :;^  • 

760211  .;,^  • 

760390  x;;  • 

.760569  hfo 
760748 i^^ • 
760927  h;Q • 

761106  2^ • 
761285  ^;!  • 
761464  ^g  • 
7616421^^- 
761821 j^^ • 
761999  U;q' 
7621771^^* 

.762356  ^^• 
762534  f- 
762712  f- 
762889  ^^• 
763067  :;'• 
763245  ;^- 
763422  :;^- 
763600  Z,' 
763777  ^^• 
763954  ^^• 

.764131  f- 
764308  ^q 
764485  ^^• 
764662  ^^• 
764838  Z,' 
765016  '^• 
765191  f- 
765367  -^• 
765544  *^- 
^^^-20  f^- 

29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29. 
29 


765<isu 
.765896 
766072 
766247 
766423 
766598 
766774 
766949 
767124 
767300 
767476 
9.767649 
767824 
767999 
768173 
768348  29 
768522  TL 
768697  if' 
768871  Zx' 
769045  ti' 
769219  "'^ 


29, 


Oosilie. 

1.913365 
913276 
913187 
913099 
913010 
912922 
912833 
912744 
912655 
912566 
912477 

9.912388 
912299 
912210 
912121 
912031 
911942 
911853 
911763 
911674 
911584 
911495 
911405 
911315 
911226 
911136 
911046 
910956 
910866 
910776 
910d86 
910596 
910506 
910415 
910325 
910235 
910144 
910054 
909963 
909873 
909782 

9.909691 
909601 
909510 
909419 
909328 
909237 
909146 
909055 
908964 
908873 
908781 
908690 
9085y9 
908607 
908416 
908324 
908233 
908141 
908049 
907968 
"sine." 


4.8 


'i'aug. 

.845227 
845496 
845764 
846033 
846302 
846670 
846839 
847107 
847376 
847644 
847913 

1.848181 
848449 
848717 
848986 
849254 
849522 
849790 
850058 
850326 
850593 

►.850861 
851129 
851396 
851664 
851931 
852199 
852466 
852733 
853001 
853268 

».  853535 
853802 
854069 
854336 
864603 
854870 

■  856137 
856404 
855671 
855938 

1.856204 
856471 
856737 
857004 
857270 
857637 
867803 
858069 
858336 
858602 

1.858868 
859134 
869400 
869666 
869932 
850198 
850464 
860730 
850995 
861261 

C!otang. 


D.  10" 

44.8 

44.8 

44.8 

44.8 

44.8 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.5 

44.5 

44.5 

44.5 

44.5 

44.5 

44.6 

44.6 

44.5 

44.5 

44 

44 

44 

44 

44 

44 

44 

44.4 

44.4 

44.4 

44.4 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 

44.3 


Cotang.     I  N.  piue.  N.  coh 


10. 


10.164773 
154504 
164236 
153967 
153698 
153430 
153161 
152893 
152624 
152356 
15S;087 

10.151819 
151651 
161283 
151014 
150746 
150478 
150210 
149942 
149675 
149407 
149139 
148871 
148604 
148336 
14«069 
147801 
147534 
147267 
146999 
146732 

10.146465 
146198 
145931 
145664 
145397 
145130 
144863 
144696 
144329 
144062 

10.143796 
143529 
143263 
142996 
142730 
142463 
142197 
141931 
141664 
141398 

10-141132 
140866 
140600 
140334 
140068 
139802 
139536 
139270 
139006 
138739 


I  67358 
57381 
'167405 
1167429 
57453 
57477 
57601 
57624 
57648 
67572 
57596 
57619 
67643 


81915 
81899 
81882 
81865 
81848 
81832 
81815 
81798 
81782 
81765 
81748 
81731 
81714 


57667  81698 


57691 
57715 
57738 
57762 
57786 
57810 


81681 
81664 
81647 
81631 
81614 
81597 


67904 
67928 
67952 
67976 
57999 
58023 
68047 
58070 


68141 
68165 


58212 


58260 
58283 
58307 


58354 
58378 
68401 
68426 
58449 
58472 
58496 
58519 
68543 
58567 


67833  81580 
57867  81563 
57881  81546 


81530 
81513 
81496 
81479 
81462 
81445 
81428 
81412 


58094,81395 
68118  81378 


81361 
81344 


58189  81327 


81310 


58236  81293 


81276 
81259 
81242 


5833081225 


81208 
81191 
81174 
81167 
81140 
81123 
81106 
81089 
81072 
81055 


58614 
58637 
58661 
68684 
68708 
58731 


58779 


Tang.      I  N.  cos.  N.si 


5869081038 


81021 
81004 
80987 
80970 
80953 
80:^36 


58755  80919 


80902 


54  Degrees. 


TABLE  II.  Ix)g.  Sines  and  Tangents.    (36°)    Natural  Sines. 


57 


Sine. 

769219 
769393 
769566 
769740 
769913 
770087 
770260 
770433 
770608 
770779 
770952 
771125 
771298 
771470 
771643 
771816 
771987 
772159 
772331 
772503 
772675 
772847 
773018 
773190 
773361 
773633 
773704 
773875 
774046 
774217 
774388 
9.774558 
774729 
774899 
775070 
775240 
776410 
775580 
775750 
775920 
776090 
776259 
776429 
776598 
776768 
776937 
777106 
777^75 
777444 
777613 
777781 
777950 
778119 
778287 
778455 
778624 
778792 
778960 
779128 
779295 
779463 


Cosine. 


D.  10'' 


Co«ine. 

.907958 
907866 
907774 
907682 
907690 
907498 
907406 
907314 
907222 
907129 
907037 

1.906945 
906852 
906760 
906667 
906576 
906482 
906389 
906296 
906204 
906111 

.906018 
905925 
905832 
905739 
905645 
905552 
905459 
905366 
905272 
905179 
9.905085. 
904992 
904898 
904804 
904711 
904617 
904523 
904429 
904335 
904241 
904147 
904053 
903959 
903864 
903770 
903676 
903581 
903487 
903392 
903298 
903202 
903108 
903014 
902919 
902824 
a02729 
902634 
902539 
902444 
902349 


Sine. 


Tang. 

.861261 
8()1627 
861792 
862058 
862323 
862589 

■  862864 
863119 
863385 
863650 
863915 

.864180 
864445 
864710 
864975 
866240 
865506 
865770 
866036 
866300 
866664 

.866829 
867094 
867368 
867623 
867887 
868162 
868416 
868680 
868945 
869209 

.869473 
869737 

,870001 
870265 
870629 
870793 
871067 
871321 
871585 
871849 

.872112 
872376 
872640 
872903 
873167 
&73430 
873694 
878967 
874220 
874484 

.874747 
876010 
87&273 
875536 
875800 
876063 
876326 
876589 
876861 
877114 


Cotang 


Cotang.  I  N.  sine.  N.  cos 


58779  80902 
58802  80885 
58826  80867 
B8849  80860 


58873 
58896 
58920 
68943 
58967 


80833 
80816 
80799 
80782 
80765 


6899080748 


59412 
59436 
59459 
69482 
59606 
69529 
59552 
! 59576 


5931880507 
69342  80489 
6936580472 
5938980455 


80438 
80422 
80403 
80386 
80368 
80351 
80334 
80316 
80299 
80282 
80264 
80247 
80230 
80212 
80195 


69599 
1 59622 

59646 

5^669 

59693 

59716 

59739 

59763  80178 
J59786f80l60 
159809^ 

69832 
159856 
i  59879 
159902 


1257801159926 


80143 
80125 
80108 
80091 
80073 
80066 


126516!!  69949  80038 


10. 


Tang. 


'60065 
i 60089 
60112 
60136 
60158 
60182 


79961 
79934 
79916 
79899 
79881 
79864 
N.j<ine, 


53  Degrees. 


58 


Log.  Sines  and  Tangents.    (37°)    Natural  Sinea. 


TABLE  n. 


Sine. 

9.779463 
779631 
779798 
779966 
780133 
780300 
780467 
780634 
780801 
780968 
781134 

9.781301 
781468 
781634 
781800 
781966 
782132 
782298 
782464 
782630 
782796 

9.782961 
783127 
783282 
783458 
783623 
783788 
783953 
784118 
784282 
784447 

9.784612 
784776 
784941 
785105 
785269 
785433 
785597 
785761 
785926 
786089 

9.786252 
786416 
786579 
786742 
786906 
787069 
787232 
787396 
787557 
787720 


D.  10" 


51  9.787883 


788046 
788208 
788370 
788532 
788694 
788856 
789018 
789180 
789342 


Cosine. 


Cosine.  ID.  10"!  Tang. 


.902349 
902253 
902158 
90-2063 
901967 
901872 
901776 
901681 
901585 
9014y0 
901394 
901298 
901202 
901106 
901010 
900914 
900818 
900722 
900626 
900529 
900433 

9.900337 
900242 
900144 
900047 
899951 
899854 
899757 
899660 
899504 
899467 

9,899370 
899273 
899176 
899078 
898981 
898884 
898787 
898689 
898592 
898494 

9.898397 
898299 
898202 
898104 
898006 
897908 
897810 
897712 
897614 
897516 
897418 
897320 
897222 
897123 
897025 
896926 
896828 
8%729 
896631 
896532 

1   Sine. 


9.877114 
877377 
877640 
877903 
878165 
878428 
878691 
878953 
879216 
879478 
879741 
880003 
880265 
880528 
880790 
881052 
881314 
881576 
881839 
882101 
882363 
882625 
882887 
883148 
883410 
883672 
883934 
884196 
884457 
884719 
884980 

9.885242 
885303 
885765 
886026 
886288 
886549 
886810 
887072 
887333 
887594 

).  887855 
888116 
888377 
888639 
888900 
889160 
889421 
889682 
889943 
890204 

1.890465 
890725 
890986 
891247 
891507 
891768 
892028 
892289 
892549 
892810 

Co  tang. 


D.  10" 


43.8 

43.8 

43.8 

43*.8 

43.8 

43.8 

43.8 

43.7 

43.7 

43.7 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43.6 

43,6 

43.6 

43.6 

43.6 

43.6 

43.5 

43.6 

43.5 

43.5 

43.5 

43.5 

43.5 

43.6 

43.5 

43.5 

43.6 

43.5 

43.5 

43.5 

43.4 

43.4 

43,4 


43.4 


Co  tang. 

10.122886 
122623 
122360 
122097 
121835 
121572 
121309 
121047 
120784 
120522 
120259 

10.119997 
119735 
119472 
119210 
118948 
118686 
118424 
118161 
117899 
117637 

10.117375 
117113 
116852 
116590 
1163281 
116066  I 
1158041 
115543 
115281  I 


116020 
10.114758 
114497 
114235 
113974 


N.sine.  N.  cos. 

60182 
60205 
60228 
60261 
60274 
60298 
60321 
60344 
60367 
60390 
60414 
60437 
60460 
60483 
G0506 
60529 
60553 
60576 
60599 
60622 
60645 
60668 
60691 
60714 
60738 
60761 
60784 
60807 
60830 
60853 


60876 
60899 
60922 
60945 
60968 


113712  ll  60991 
113451  61015 
113190  !;  61038 


79864 
79846 
79829 
79811 
79793 
79776 
79758 
79741 
79723 
79706 
79688 
79671 
79658 
79635 
79618 
79600 
79583 
79565 
79547 
79530 
79512 
79494 
79477 
79469 
79441 
79424 
79406 
79388 
79371 
79353 
79335' 
79318 
79300 
79282 
79264 
79247 
9229 
79211 


112928  i  61061  79193 


112667  1  61084 
112406  116110 


10.112145  i!61130 79140 


111884 


61153 


61176 
61199 
61 22-. 


111623 
111361 
111100 
110840 '161245 
110579;  161268 
110318  161291 
110057  161314 
109796  161337 
10.109533  161360 
109275:161383 
109014,161406 
108753  '61429 
108493  161451 
108232  161474 
107972  ;|61497 
107711  !|61520 
1074511  61543 
1071901!  61566 


Tang. 


II  N.  co«.  N.sine 


79176 
79158 


9122 
79105 
79087 
79069 
79051 
79033 
?9016 
;8998 
78980 
78962 
78944 
78926 
78908 
78891 
78873 
78855 

8837 
78819 
78801 


52  Degrees. 


60 


Log.  Sinos  and  Tangents.    (39°)    Natural  Sines. 


TABLE  n. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

H 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 


9.798772 
799028 
799184 
799339 
799495 
799651 
799806 
799962 
800117 
800272 
800427 

9.800582 
800737 
800892 
801047 
801201 
801356 
801511 
801665 
801819 
801973 

J. 802128 


D.  10 


802282 
802436 
802689 
802743 
8028^7 
803050 
803204 
803367 
803611 
9.803664 
803817 
803970 
804123 
804276 
804428 
804581 
804734 
804886 
805039 
805191 
805343 
805495 
805647 
805799 
806951 
806103 
806254 
806406 
806557 
9.806709 
806860 
807011 
807163 
807314 
807466 
807615 
807766 
807917 
808067 
Cosine. 


26.0 

26.0 

26.0 

25.9 

25.9 

25.9 

26.9 

25.9 

25.9 

25.8 

25.8 

25.8 

25.8 

25.8 

25.8 

25.8 

25.7 

25.7 

25.7 

25.7 

25.7 

25.7 

25.6 

25.6 

25.6 

25.6 

25.6 

25.6 

25.6 

25.5 

25.5 

25.5 

25.5 

25.5 

25.5 

25.4 

25.4 

26.4 

25.4 

25.4 

25.4 


CxMiine. 


25 

25 

25 

25 

25 

25 

25 

25.3 

25.2 

25.2 

25.2 

25.2 

25.2 

25.2' 

25.2 

25.1 

25.1 

25.1 

25.1 


9.890503 
890400 
890298 
890195 
890093 
889990 
889888 
889785 
889682 
889579 
889477 
9.889374 
889271 
889168 
889064 
888961 
888858 
888755 
888651 
8385'18 
8SH444 
). 888341 
888237 
888134 
888030 
887926 . 
887822 
887718 
837614 
887510 
887406 
>.  887302 
887198 
887093 
886989 
88G885 
886780 
886676 
886571 
886466 
886362 
9.886257 
886152 
886047 
885942 
885837' 
885732 
885627 
885622 
885416 
885311 
9.885205 
885100 
884994 
884889 
884783 
884677 
884572 
884466 
884360 
884264 


D.  10" 


Sine. 


17.0 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.4 

17. .4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.5 

17.5 

17.6 

17.5 

17.6 

17.5 

17.6 

17.5 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 

17.6 


Tang. 


9.908369 
90S628 
90S886 
909144 
909402 
903660 
909918 
910177 
910435 
910693 
910951 
9.911209 
911467 
911724 
911982 
912240 
912498 
912766 
913014 
913271 
913529 
9.913787 
914044 
914302 
914660 
914817 
916075 
915332 
916590 
916847 
916104 
9.916362 
916619 
916877 
917134 
917391 
917648 
917905 
918163 
918420 
918677 
9.918934 
919191 
919448 
919705 
919962 
920219 
920476 
920733 
920990 
921247 
). 921503 
921760 
922017 
922274 
922630 
922787 
923044 
923300 
923657 
923813 
Cotang. 


D.  10" 


43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42,9- 

42,9 

42.9 

42.9 

42,9 

42.9 

42.9 

42,9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.7 


Cotaug. 


10.091631 
091372 
091114 
090856 
090598 
090340 
090082 

089823 

089505 

089307 

089049 
10.088791 

0S8533 

088276 

088018 

087760] 

087502 

087244 ! 

a8S986 

086729 

086471 1 
iO. 036213 

036956 

085698 

085440 

035183 

034925 

084668 

084410 

084163 

033896 
10.083638 

083381 

083123 

082866 

082609 

082352 

082096 

081837 

081680 

081323 
10-081066 
080809 
080552 
030295 
080038 
079781 
079524 
07926? 

079010  It  64033 
078753 j  1 64056 
10.0784971164078 
07824011  64100 
077983  |i6412G 
077726 

077470  !i  64167 
077213;;  64190 
0769561164212 
076700 1  j  64234 
07644311 64266 
0761871164279 


N.  sine.  N.  cos. 

77716 

77696 

77678 

77660 

77641 

77623 

77605 

77586 

77668 

77550 

77531 

77513 

77494 

77476 

77458 

77439 

77421 

77402 

77384 

77366 

77347 

77329 

77310 

77292 

77273 

77255 

77236 

77218 

77199 

77181 

77162 

77144 

77125 

77107 

77088 

77070 

77051 

77033 

77014 

76996 

76977 

76959 


62932 

62955 

62977 

63000 

63022 

63045 

63068 

63090 

63113 

63135 

63158 

93180 

63203 

63225 

63248 

63271 

63293 

63316 

63338 

63361 

63383 

63406 

63428 

63451 

63473 

63496 

63518 

63540 

63663 

63585 

63608 

63630 

63653 

63675 

63698 

63720 

63742 

63765 

63787 

63810 

63832 

63854 

63877  76940 

63899  76921 


6392: 


60 

59 

58 

57 

66 

65 

64 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33  ! 

32 

31 

30 

29 

28 

27 

26 

25 

24 


76903 


63944 176884 

j  63966  76866 

1 6398:.'  76847 

164011,76828 

76810 

76791 

76772 

76754 

76736 

64145  76717 

76098 

6679 

76661 

76642 

76623 

76604 


Tang.  U  N.  pou.  N.Bine 


50  Degrees. 


TABLE  n. 


Log.  SineB  and  Tangents.    (40*)    Natural  Sines. 


61 


dTio^ 


Sine. 


D.  IC 


810017 
810167 
810316 
810465 
810614 
810763 
810912 
811061 

.811210 
811358 
811507 
811655 
811804 
811952 
812100 
812248 
812396 
812544 

.812692 
812840 
812988 
813135 
813283 
813430 
813578 
813725 
813872 
814019 

.814166 
814313 
814460 
814607 
814753 
814900 
815046 
815193 
815339 
815485 

1.816631 
815778 
815924 
816069 
816215 
816361 
816507 
816652 
816798 
816943 
Cosine. 


25.1 
25.1 
25.1 
25.0 
25.0 
25.0 
25.0 
25.0 
26.0 
24.9 
24.9 
24.9 
24.9 
24.9 
24.9 
24.8 
24.8 
24.8 
24.8 
24.8 
24.8 
24.8 
24.7 
24.7 
24.7 
24.7 
24.7 
24.7 
24.7 
24.6 
24.6 
24.6 
24.6 
24.6 
24.6 
24.6 
24.5 
24.6 
24.5 
24.5 
24.5 
24.5 
24.5 
24.4 
24.4 
24.4 


Coaiae.  jD.  10" 


24 

24 

24 

24 

24 

24 

24,3 

24,3 

24.3 

24.3 

24.3 

24.2 

24.2 

24.2 


883723 
883617 
883510 
883404 
883297 
883191 

.883084 
882977 
882871 
882764 
882667 
882550 
882443 
882336 
882229 
882121 

.882014 
881907 
881799 
881692 
881584 
881477 
881369 
881261 
881153 
881046 

. 880938- 
880830 
880722 
880613 
880505 
880397 
880289 
880180 
880072 
879963 

.879855 
879746 
879637 
879529 
879420 
879311 
879202 
879093 
878984 
878875 

.878766 
878656 
878547 
878438 
878328 
878219 
878.09 
877999 
877890 
877780 
Sine. 


8.0 

8.0 

8.0 

8.0 

8.0 

8.0 

8.0 

8.0 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8. 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.3 

8.3 

8.3 

8.3 


Tang. 


1.923813 
924070 
924327 
924583 
924840 
925096 
925352 
925609 
926865 
926122 
926378 

I.926G34 
926890 
927147 
927403 
927669 
927915 
928171 
928427 
928683 
928940 

.929196 
929452 
929708 
929964 
930220 
930475 
930731 
930987 
931243 
931499 

'.931755 
932010 
932266 
932522 
932778 
933033 
933289 
933545 
933800 
934056 

.934311 
934567 
934823 
935078 
935333 
9365S0 
535844 
936100 
936356 
936610 

1.936866 
937121 
937376 
937632 
937887 
938142 
938398 
938653 
938908 
939163 

Cotang. 


Cotang.  i  N  .«ine.  N.  cos. 


10.076187 
076930 
076673 
076417 
075160 
074904 
074648 
074391 
074135 
073878 
073622 

10.073366 
073110 
072863 
072597 
072341 
072085 
071829 
071573 
071317 
071060 

10.070804 
070548 
070292 
070036 
069780 
069526 
069269 
069013 
068767 
068601 

10.068245 
0679901 
067734  t 
067478  I 
067222  I 
066967 
066711 
066455 
066200 
065944 

10.065689 

065433 

065177 

064922  i 

064667  I 

064411  I 

064166  i 

063900 ! 

063646  i 

063390  I 

10.0631341 
062879  I 
062624  I 
062368 
062113 
061868 
061602 
061347 
061092 
060837 


64279 
64301 
64323 
64346 
64368 
64390 
64412. 
64436 
64457 
64479 
64501 
64624 
64646 
64668 
64690 
64612 
64636 
64657 
64679 
64701 
64723 
64746 
'64768 
64790 
64812 
64834 
64856 
64878 
64901 
64923 
64945 
64967 
64989 
65011 
65033 
65065 
65077 
65100 
65122 
65144 
65166 
65188 
65210 
65232 
65264 
66276 
66298 
65320 
65342 
65364 
65386 
65408 
65430 
65452 
66474 
65496 
65518 
65540 
65562 
66584 
66606 


Tang. 


76604 
76686 
76567 
76648 
76630 
76611 
76492 
76473 
76466 
76436 
76417 
76398 
6380 
76361 
76342 
76323 
76304 
76286 
76267 
76248 
76229 
76210 
76192 
76173 
76164 
76135 
76116 
76097 
76078 
76059 
76041 
76022 
76003 
76984 
75965 
75946 
76927 
76908 
76889 
76870 
76861 
75832 
75813 
75794 
76775 
75766 
75738 
76719 
75700 
756S0 
75661 
76642 
75623 
75604 
75585 
75666 
75547 
75528 
75609 
76490 
76471 


N.  COS.  N.sine. 


49  Degrees. 


Log.  Sines  and  Tangents.    (41°)    Natural  Sines. 


TABLE  IL 


Sine.       D.  10"     C!osiue 


9.816948 
817088 
817233 
817379 
817524 
817668 
817813 
817958 
818103 
818247 
818392 

9.81S636 
818681 
818825 
818969 
819113 
819257 
819401 
819545 
819689 
819832 

9.819976 
820120 
820263 
820405 
820550 
820693 
820836 
820979 
821122 
821265 

9.821407 
821550 
821693 
821835 
821977 
822120 
822262 
822404 
822546 
822688 

9.822830 
822972 
823114 
823256 
823397 
823639 
823680 
823821 
823963 
824104 

9.824245 
824386 
824627 
82-1668 
82-1808 
824949 
826090 
825230 
825371 
825511 
Cosine. 


24.2 
24.2 
24.2 
24.2 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.1 
24.0 
24.0 
24. 0 
24.0 
24.0 
24.0 
24.0 
23.9 
23.9 
23.9 
23.9 

23.  y 

23.9 
23.9 
23.8 
23.8 
23.8 
23.8 
23.8 
23.8 


23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23.6 

23.6 

23.6 

23.6 

23.5 

23.6 

23.5 

23.6 

23.6 

23.6 

23.6 

23.4 

23.4 

23.4 

23.4 

23.4 

23.4 


.877780 
877670 
877560 
877450 
877340 
877230 
877120 
877010 
876899 
876789 
87667 
.876568 
876457 
876347 
876236 
876125 
876014 
876904 
875793 
876682 
876571 
,876469 
876348 
876237 
876126 
876014 
874903 
874791 
874680 
874568 
874456 
,874344 
874232 
874121 
874009 
873896 
873784 
873672 
873660 
873448 
873336 
873223 
873110 
872998 
872886 
872772 
872659 
872547 
872434 
872321 
872208 
872095 
871981 
871868 
871755 
871641 
871528 
871414 
871301 
871187 
871073 
"si]ie.~ 


9.87 


D.  10" 


18.4 


18 

18 

18 

18 

18 

18 

18 

18 

18 

18 

18 

18,6 

18.6 

18.6 

18.6 

18.6 

18.6 

18. & 

18.6 

18.6 

18.7 

18.7 

18.7 

18,7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.7 

18.8 

18.8 

IS. 8 

18.8 

18.8 

18.8 

18.8 

18.8 

18.3 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 

18.9 


'I'ang. 

9.939163 
939418 
939673 
939928 
940183 
940438 
940694 
940949 
941204 
941458 
941714 

9.941968 
942223 
942478 
942733 
942988 
943243 
943498 
943752 
944007 
944262 

9.944517 
944771 
945026 
945281 
945535 
945790 
946045 
946299 
946554 
946808 

9.947053 
947318 
947572 
947826 
948081 
948336 
948590 
948844 
949099 
949353 

9.949607 
949862 
950116 
950370 
950625 
950879 
951133 
951388 
951642 
951896 

3.952150 
952405 
952659 
952913 
953167 
953421 
953675 
963929 
954183 
964437 
Co  tang. 


D.  10" 


42.5 

42.6 

42.5 

42.6 

42.5 

42.6 

42 

42 

42 

42 

42 

42 

42 

42 

42.5 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.6 

42.4 

42.4 

42.4 

42 

42 

42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42 


Cotang.  I  iN.  sine. 


42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.3 

42.3 

42.3 

42.3 

42.3 


10.030837  i!  66606 
060582 j  66628 
060327  165650 
060072  I  j 65672 
059817  !JG5694 
05956-2  i  I  65716 
059306  jj  65738 
059051  65759 
058796  1 1  65781 
0585421165803 
0582S6!j65825 

10.0580321166847 
057777  1 1 65869 
067622!  1 65891 


057267 
057012 
056757 
056502 
056248 
055993 
055738 
10.055483 
055229 


66913 
65935 
65966 
65978 
66000 
66022 
66044 
66066 
66088 


054974  1 1 66109 
0547191166131 
054465  166153 


054210 
053965 
053701 
063446 
053192 
10.062937 
062682 
052428 


i 66176 
166197 
[66218 
166240 
1 66262 
66284 
1 66306 
166327 


052174 j  66349 
06191911 66371 
051664 1 1 66393 
0514101  66414 
051166  66436 
050901  66458 
050647  I  66480 

10  050393  66601 
050138  I  j 66623 
049884 1166645 
049630 
049376 
049121 
048867 1  66632 
0486121 1 66653 
048358  166676 
048104  j 66697 

10.047850 1  66718 
047595  1 1 66740 
047341]  166762 
047087  '66783 


66666 
C6688 
66610 


046833 
046679 
046325 
046071 
045817 
045563 


66805 
66827 
66848 
66870 
66891 
66913 


75471 

76462 
?5433 
75414 
76396 
75375 
76366 
76337 
75318 
76299 
75280 
75261 
76241 
75222 
76203 
75184 
75166 
76146 
76126 
75107 
76088 
76069 
76050 
75030 
76011 
74992 
74973 
74963 
74934 
74915 
74896 
74876 
74857 
74838 
74818 
74799 
74780 
74760 
74741 
74722 
74703 
74683 
74663 
74644 
74625 
74606 
74586 
74667 
74548 
74622 
74509 
74489 
74470 
4461 
74431 
74412 
74392 
74373 
74353 
4334 
74314 


N.  COS.  N.sine 


48  Degrees. 


TABLE  n. 


Log.  Sines  and  Tangents.  (42°)*  Natural  gines. 


63 


Sine. 


D.  10"  Cosine. 


D.  10" 


Tang. 


D.  10' 


Cotang.  ;N.  Bine. IN.  COB 


.825.^11 
826651 
825791 
825931 
826071 
826211 
826351 
826491 
826631 
826770 
826910 

.827049 
827189 
827328 
827467 
827606 
827745 
827884 
828023 
828162 
828301 

.828439 
828578 
828716 
828S55 
828993 
829131 
829269 
829407 
829545 
829683 

.829821 
829959 
830097 
830234 
830372 
830509 
830646 
830784 
830921 
831058 

.831195 
831332 
831469 
831606 
831742 
831879 ! 
832015 
832162 
882288 
832425 
832561 
832697 
832833 
832969 
833105 
833241 
833377 
833612 
833648 
833783 


23.4 
23.3 
23.3 
23.3 
23.3 
23.3 
33.3 
23.3 
23.3 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.7 
23.7 
22.7 

:22.7 

122.7 

122.7 

1 22.7 

'22.6 

!22.6 

122.6 

122.6 

^22 

1 22.6 


Co.siiie. 


.871073 
870960 
870846 
870732 
870618 
87050-i 
870390 
870276 
870161 
870047 
869933 
.869818 
869704 
869589 
869474 
869360 
869246 
869130 
869015 
868900 
868785 
.868670 
868555 
868440 
868324 
868209 
868093 
867978 
867862 
867747 
867631 
.867515 
867399 
867283 
867167 
867051 
866935 
866819 
866703 
866586 
866470 
.866853 
866237 
866120 
866004 
865887 
865770 
865653 
865536 
865419 
865302 
1.865185 
865068 
864950 
864833 
864716 
864598 
864481 
864363 
864245 
864127 


19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.1 
19.4 
19.5 
19.5 
19.6 
19.6 
19.5 
19.5 
19.5 
19.5 
19.5 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 


.964437 
964691 
954945 
955200 
955454 
955707 
955961 
956215 
956469 
956723 
956977 

.957231 
957485 
957739 
957993 
958246 
958500 
958764 
959008 
959262 
959516 

.959769 
960023 
960277 
960531 
960784 
961038 
961291 
961545 
961799 
962052 

.962306 
962560 
962813 
963067 
963320 
963574 
963827 
964081 
964335 
964588 

.9t>4J542 
9(i5095 
965349 
965602 
965856 
966109 
966362 
966616 


967123 
.9S7376 
967629 
967883 
968136 
968389 
968643 
968896 
969149 
969403 
969656 


42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
43.3 
42.3 
43.3 
42.3 
43.8 
43.3 
43.3 
43.3 
43.3 
43.3 
43.3 
43.3 
42.8 
42.3 
43.3 
43.3 
43.3 
43.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 
42.3 


42 

42 

42 

42 

42 

42 

42 

42 

42 

43.2 

42.2 

42.2 

42.3 

42.2 

43.3 

42.2 

42.2 

42.3 

43.2 

43.3 

43.2 

42.2 

42.2 

42.2 


10.045563 
045309 
045065 
044800 
044546 
044293 
044039 
043785 
043531 
043277 
043023 

10.042769 
042615 
042261 
042007 
041764 
041500 
041246 
040992 
040738 
040484 

10.040231 
039977 
039723 
039469 
039216 
038962 
038709 
038466 
038201 
037948 

10.037694 
037440 
037187 
036933 
036680 
036426 
036173 
035919 
035665 
035412 

10.036168 
034905 
034651 
034398 
034145 
033891 
033638 
033384 
033131 
032877 

10.032624 
032371 
032117 
031864 
031611 
031357 
031104 
030851 
030597 
030344 


||66913|74314 
1 1 66935174295 
j!  66956174276 
''66978174266 
i!  66999  74237 


ii  67021 
167043 
!' 67064 
'i  67086 
ii  67107 
i!  67129 
;:  67151 
!|  67172 
i  1 67194 
67215 
1 1 67237 
i!  67258 


74217 
74198 
74178 
74169 
74139 
74120 
74100 
74080 
74061 
74041 
74022 
74002 


1 67280|73983 


Cotang. 


Tang. 


3963 
3944 
73924 
73904 
73885 
73865 
73846 
73826 
73806 
73787 
3767 
73747 
73728 
73708 
73688 
73669 
73649 
73629 
73610 
73590 
73570 
73551 
73531 
73611 
73491 
73472 
73452 
73432 
73413 
73393 
373 
73353 
73333 
73314 
73294 
73274 
73254 
73234 
73216 
73195 
73175 
73155 
73136 
N.  CO?.  N.Hin«, 


1167301 

I  {67323 

167344 

i 1 67366 

167387 

167409 

67430 

!  67452 

67473 

i 167495 

i 1 67516 

167638 

1 167559 

i 1 67580 

67602 

ii  67623 

ij  67646 

i 167666 

67688 

67709 

67730 

67762 

67773 

67795 

67816 

I  67837 

! 67859 

167880 

167901 

I  67923 


67944 


1:67966 
1 1 67987 
1 1 68008 
i  68029 
ii  68051 
I  68072 
68093 
168115 
1 68136 
68157 
iG8179 
i  68200 


47  Degrees^ 


27 


64 


Log.  Sines  and  Tangenta.    (43°)    Natural  Sines. 


TABLE  n. 


Sine. 
0  9.833783 


1 
2 
3 

4 

6 

6 

7 

6 

9 

10 

U 

12 

13 

14 

16  J 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


833919 
834054 
834189 
834325 
834460 
834595 
834730 
834865 
834999 
835134 

9,835269 
835403 
835538 
835672 
835807 
835941 
836075 
836209 
836343 
836477 

9,836611 
836745 
836878 
837012 
837146 
837279 
837412 
837646 
837679 
837812 

9,837945 
838078 
838211 
838344 
838477 
838610 
838742 
838875 
839007 
839140 

9,839272 
839404 
839536 
839668 
839800 
839932 
840064 
840196 
840328 
840459 

9,840591 
840722 
840854 
840986 
841116 
841247 
841378 
841509 
841640 
841771 
Cosine. 


D.  ly^l  Cosine.  iD.  m       Tang.   D.  10"|  Cotang 


22.6 
22.5 
22.5 
22.5 
22.5 
22.5 
22.5 
22.5 
22.5 
22.4 
22.4 
22.4 
22.4 
22.4 
22,4 
22.4 
22.4 
22.3 
22.3 
22.3 
22.3 
22.3 
22.3 
22.3 
22.2 
22.2 
22.2 
22.2 
22.2 
22.2 
22.2 
22.2 
22.1 
22,1 
22.1 
22.1 
22,1 
22,1 
22.1 
22.1 
22,0 
22.0 
22,0 
22.0 
22.0 
22,0 
22,0 
9 

21,9 
21.9 
21.9 
21.9 
21.9 
21.9 
21.9 
21.8 
21.8 
21.8 
21.8 
21.8 


1.864127 
864010 
863892 
863774 
863656 
863538 
863419 
863301 
863183 
863064 
862946 
.862827 
862709 
862590 
862471 
862353 
862234 
862115 
861996 
861877 
861758 
.861638 
861519 
861400 
861280 
861161 
861041 
860922 
860802 
860682 
860662 
.860442 
860322 
860202 
860082 
869962 
859842 
859721 
859601 
869480 
859360 
859239 
859119 
858998 
858877 
868756 
868636 
868514 
868393 
858272 
858151 
,858029 
857908 
857786 
857665 
857543 
857422 
857300 
857178 
857056 
856934 
Sine. 


19.6 

19.6 

19.7 

19.7 

19.7 

19,7 

19.7 

19.7 

19.7 

19.7 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.1 

20.1 

20,1 

20.1 

20,1 

20,1 

20.1 

20.1 

20,2 

20.2 

20.2 

20,2 

20,2 

20.2 

20.2 

20.2 

20,2 

20.3 

20.3 

20.3 

20.3 

20.3 

20.3 


970162 
970416 
970669 
970922 
971175 
971429 
971682 
971935 
972188 
9.972441 


9.969656  .„  o 
969909  ^Z-i 


972948 
973201 
973454 
973707 
973960 
974213 
974466 
974719 

9.974973 
975226 
976479 
976732 
975985 
976238 
976491 
976744 
976997 
977250 
.977503 
977766 
978009 
978262 
978615 
978768 
979021 
979274 
979527 
979780 

9.980033 
980286 
980538 
980791 
981044 
981297 
981560 
981803 
982056 
982309 

9.982562 
982814 
983067 
983320 
983573 
983826 
984079 
984331 
984584 
984837 
Cotang.  1 


42.2 

42.2 

42.2 

42.2 

42,2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42,2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42,2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42,2 

42,1 

42.1 

42.1 

42.1 

42.1 

42,1 

42,1 

42.1 

42,1 

42,1 

42,1 

42,1 

42,1 

42,1 

42.1 

42.1 


I- 

10.030344 
030091 
029838 
029584 
029331 
029078 
028825 
028571 
028318 
028066 
027812 

10.027559 
027306 
027062 
026799 
026546 
026293 
026040 
025787 
025534 
025281 

10.025027 
024774 
024521 
024268 
024015 
023762 
023609 
023266 
023003 
022750 

10.022497 
022244 
021991 
021738 
021485 
021232 
020979 
•  020726 
020473 
020220 

10.019967 
019714 
019462 
019209 
018956 
018703 
018450 
018197 
017944 
017691  I 

10.017438 
017186 
016933 
016680 
016427 
016174 
015921 
015669 
015416 
015163 
"fang. 


|N  .sine.  N.  cos 


68200 
68221 
68242 
68264 
68285 
68306 
68327 
68349 
68370 
68391 
68412 
68434 
68455 
68476 
68497 
68618 
68639 
68561 
68582 
68603 
68624 
68645 
68666 


68709 
68730 
68751 
68772 
68793 
68814 
68836 
68857 
68878 
68899 
68920 
68941 
68962 


73135 
73116 
73096 
73076 
73056 
73036 
73016 
72996 
72976 
72957 
72937 
72917 
72897 
72877 
72857 
72837 
72817 
72797 
72777 
72757 
72737 
72717 
72697 
72677 
72657 
72637 
72617 
72697 
72677 
72557 
72537 
72517 
72497 
72477 
72457 
72437 
72417 


68983  72397 
6900472377 
6902672367 
69046  72337 


69067 
69088 
69109 
69130 
69151 
69172 


72317 
72297 
72277 
72257 
72236 
72216 


69193  72196 
169214172176 
69235  72156 
69256I721S6 
69277  i721 16 
69298172095 
69319  72075 
6934072055 
6936172035 
69382  72015 


69403 
69424 
69445 
69466 


71995 
71974 
71954 
71934 


N.  cos.lN.sine. 


46  Degrees. 


TABLE  IT. 


Log.  Sines  and  Tangents.    (44°)    Natural  Sines. 


65 


Sine. 

,841771 
841902 
842033 
842163 
842294 
842424 
842555 
842685 
842815 
842946 
843076 
.843206 
843336 
843466 
843595 
843725 
843855 
843984 
844114 
844243 
844372 
844502 
844631 
844760 
844889 
845018 
845147 
845276 
845405 
845533 
846662 
9.845790 
845919 
846047 
846175 
846304 
846432 
846560 
846688 
846816 
846944 

.847071 
847199 
847327 
847454 
847582 
847709 
847836 
847964 
848091 
848218 

.848346 
848472 
848599 
848726 
848852 
848979 
849106 
849232 


849486 


Cosine. 


D.  10" 


21.8 
21.8 
21.8 
21.7 
21,7 
21.7 
21.7 
21,7 
21.7 
21.7 
21.7 
21.6 
21,6 
21,6 
21.6 
21.6 
21.6 
21.6 
21.6 
21.5 
21.5 
21.5 
21.5 
21.5 
21.5 
21.5 
21.6 
21.4 
21.4 
21.4 
21.4 
21,4 
21.4 
21.4 
21.4 
21.4 
21.3 
21.3 
21.3 
21.3 
21.3 
21.3 
21.3 
21,3 
21.2 
21,2 
21.2 
21,2 
21,2 
21.2 
21.2 
21.2 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 


Cosine. 


9.866934 
866812 
866690 
856568 
856446 
866323 
866201 
866078 
855956 
866833 
855711 
855588 
855465 
855342 
865219 
855096 
854973 
854850 
854727 
854603 
854480 

9.854356 
854233 
854109 
853986 
853862 
863738 
853614 
853490 
853366 
853242 
853118 
852994 
852869 
862745 
862620 
852496 
852371 
852247 
852122 
861997 

9.851872 
851747 
851622 
851497 
851372 
851246 
851121 
850996 
850870 
860745 
850619 
850493 
850368 
850242 
850116 
849990 
849864 
849738 
849611 
849485 


Sine. 


D.10" 


20.3 
20.3 
20.4 
20.4 
20.4 
20.4 
20.4 
20.4 
20.4 
20.4 
20.6 
20.6 


20 

20 

20 

20 

20 

20 

20.6 

20.6 

20,6 

20.6 

20.6 

20,6 

20.6 

20.6 

20.6 

20.7 

20.7 

20,7 

20.7 

20.7 

20,7 

20.7 

20.7 

20.7 

20.8 

20,8 

20,8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.9 

20.9 

20,9 

20.9 

20,9 

20,9 

20,9 

20.9 

21,0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 


Tang. 


984837 
985090 
985343 
985696 
985848 
986101 
980354 
986607 
986860 
987112 
987365 
987618 
987871 
988123 
988376 
988629 
988882 
989134 
989387 
989640 
989893 

9.990146 
990398 
990661 
990903 
991156 
991409 
991662 
991914 
992167 
992420 
992672 
992926 
993178 
993430 
993683 
993936 
994189 
994441 
994G94 
994947 

9.995199 
996452 
995705 
995967 
996210 
996463 
996716 
996968 
997221 
997473 
,997726 
997979 
998231 
998484 
998737 
998989 
999242 
999495 
999748 

10.000000 
Co  tang. 


D.  10" 


Cotang. 


N.  sine 


10.015163 
014910 
014667 
014404 
014162 
013899 
013646 
013393 
013140 
013888 
012636 

10.012382 
012129 
011877 
011624 
011371 
011118 
010866,1 
0106131 
010360,' 
010107  i 

10.009855  ii 
009602 
009349 
009097 
008844 
608691 
008338 
008086 
007833 
007580 

10-007328 
007075 
006822 
006570 
006317 
006064 
005811 i 
005559 
005306 
005063 

10.004801 
004648 
004296 
004043 
003790 
003537 
003285 
003032 
002779  i 
002527 ! 

10.002274  I 
002021 
001769 
001516 
001263 
001011 
000758 
000505 
000263 
000000 


69466 
69487 
69508 
69529 
69549 
69570 
69591 
69612 


69633  71772 
69654  71752 


Tang. 


69675 
69696 
69717 
69737 
69758 
69779 
69800 
69821 
69842 
698G2 
69883 
69904 
69925 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 
70112 
70132 
70153 
70174 
70196 
70215 
70236 
70257 
70277 
70298 
70319 
70339 
70360 
70381 
70401 
70422 
70443 
70463 
70484 
70505 
70525 
70546 
70567 
70587 
70608 
70628 
70649 
70670 
70691 


N.  cos 


71934 
71914 
71894 
71873 
71853 
71833 
71813 
71792 


71732 
71711 
71691 
71671 
71650 
71630 
71610 
71590 
71569 
71549 
71529 
71508 
71488 
71468 
71447 
71427 
71407 
71386 
71366 
71345 
71325 
71305 
71284 
71264 
71243 
71223 
71203 
71182 
71162 
71141 
71121 
71100 
71080 
71059 
71039 
71019 
70998 
70978 
70957 
0937 
70916 
70896 
70876 
70856 
70834 
70813 
70793 
70772 
70752 
70731 


7071170711 


N.  COS.  N. mm- 


45  Degrees. 


66 

LOGARITHMS 

TABLE  HI 

. 

LOGARITHMS    OF 

NUMBERS, 

FROM  1  TO  110, 

INCLUDING 

TWELVE  DECIMAL  PLACES 

N. 
I 

Log. 

N. 

36 

Log. 

0.    000 

000 

000  000 

1.    656 

302 

500 

767 

3 

0.    301 

029 

995  644 

37 

I.        568 

201 

724 

067 

3 

0.    477 

121 

254  720 

38 

1.    579 

783 

596 

617 

4 

0.    602 

059 

991  328 

39 

1.    591 

064 

607 

264 

6 

0.    698 

970 

004  336 

40 

1.    602 

059 

991 

328 

6 

0.    778 

151 

250  384 

41 

1.    612 

783 

846 

720 

7 

0.    845 

098 

040  014 

42 

I,    623 

249 

290 

398 

8 

0.    903 

089 

986  992 

43 

1.    633 

468 

465 

679 

9 

0.    954 

242 

509  440 

44 

1.    643 

452 

676 

486 

10 

1,    000 

000 

000  OOO 

45 

U    663 

212 

513 

775 

H 

1.    041 

392 

685  158 

46 

1.    662 

767 

831 

682 

12 

1.    079 

181 

246  048 

47 

1.    672 

097 

857 

936 

13 

1,    113 

943 

352  309 

48 

1.    681 

241 

237 

376 

14 

1.    146 

128 

035  678 

49 

1.    690 

196 

080 

028 

15 

1.    176 

091 

259  059 

60 

1.    698 

970 

004 

336 

16 

1.    204 

119 

982  656 

51 

1,        707 

570 

176 

098 

17 

1,    230 

448 

921  378 

52 

1.    716 

003 

243 

635 

18 

1.    255 

272 

505  103 

53 

1.    724 

275 

869 

601 

19 

1.    278 

753 

600  953 

54 

I.    732 

393 

769 

823 

20 

1,    301 

029 

995  664 

55 

1.    740 

362 

689 

494 

21 

1,    322 

219 

294  734 

56 

1    748 

188 

027 

006 

22 

1.    342 

422 

680  822 

67 

1.    756 

874 

855 

672 

23 

1.    361 

727 

836  076 

58 

1.    763 

427 

993 

663 

24 

1.    380 

211 

241  712 

69 

1.    770 

852 

Oil 

642 

25 

1.    397 

940 

008  672 

60 

1.    778 

161 

250 

384 

26 

1.    414 

973 

347  971 

61 

I.    785 

329 

835 

Oil 

27 

1.    431 

363 

764  159 

62 

1.    792 

391 

689 

492 

28 

1.    447 

158 

031  342 

63 

1.'   799 

340 

649 

464 

29 

1.    462 

397 

997  899 

64 

1,    806 

179 

973 

984 

30 

1.    477 

121 

254  720 

65 

1.    812 

913 

366 

643 

31 

1.    491 

361 

693  834 

66 

1.    819 

543 

935 

542 

32 

1.    505 

149 

978  320 

67 

1.    826 

074 

302 

701 

33 

1.    618 

513 

939  878 

68 

1.    832 

608 

912 

706 

34 

1.    531 

478 

917  042 

69 

1.    838 

849 

090 

737 

35 

1.    544 

068 

044  350 

70 

1.    845 

098 

040 

014 

OF  NUMBERS. 

67 

■  N. 

Log. 

N. 

Log. 

71 

851 

258 

348 

719 

91 

959 

041 

392 

321 

72 

857 

332 

496 

431 

92 

968 

787 

827 

346 

73 

863 

322 

860 

120 

93 

968 

482 

948 

654 

74 

869 

231 

719 

731 

94 

973 

127 

853 

600 

75 

875 

om 

263 

392 

95 

977 

723 

605 

289 

76 

880 

813 

592 

281 

96 

982 

271 

233 

040 

77 

886 

490 

725 

172 

97 

986 

771 

734 

266 

78 

892 

094 

602 

690 

98 

991 

226 

075 

692 

79 

897 

627 

091 

290 

99 

995 

635 

194 

59S 

80 

903 

089 

986 

992 

100 

2. 

000 

000 

000 

000 

81 

908 

485 

018 

879 

101 

2, 

004 

321 

373 

783 

82 

9i3 

813 

852 

384 

102 

2. 

008 

600 

171 

762 

83 

9i9 

078 

092 

376 

103 

2. 

012 

837 

224 

705 

84 

924 

279 

286 

062 

104 

2. 

017 

033 

339 

299 

85 

929 

418 

925 

714 

105 

2. 

021 

189 

299 

070 

8G 

934 

498 

451 

244 

106 

2. 

025 

305 

865 

265 

87 

939 

519 

252 

619 

107 

2. 

029 

383 

777 

685 

88 

944 

482 

672 

150 

108 

2. 

033 

423 

755 

487 

89 

949 

390 

006 

645 

109 

•2. 

037 

426 

497 

941 

90 

954 

242 

509 

439 

110 

2. 

041 

392 

685 

158 

LO 

GARITHMS 

OF  THE  PRIME  NUMBERS 

FROM  IK 

)  TO  ] 

11^9. 

I 

NCLUDING 

TWELV] 

K  DE 

CIMAL  PLACES 

N. 

Log. 

N. 

Log. 

113 

2. 

053 

078 

443 

483 

197 

2. 

294 

466 

266 

162 

127 

2. 

103 

803 

720 

956 

199 

2. 

298 

853 

076 

410 

131 

2. 

117 

271 

295 

656 

211 

2. 

324 

282 

455 

298 

137 

2. 

136 

720 

567 

156 

223 

2. 

348 

304 

863 

222 

139 

2. 

143 

014 

8G0 

254 

227 

2. 

356 

025 

857 

189 

149 

3. 

173 

186 

268 

412 

229 

2. 

359 

835 

482 

343 

151 

2. 

178 

976 

947 

293 

233 

2. 

367 

355 

922 

471 

157 

2. 

195 

899 

653 

409 

239 

2. 

378 

397 

902 

352 

163 

2. 

212 

187 

604 

404 

241 

2. 

382 

017 

042 

576 

167 

2. 

222 

716 

471 

148 

251 

2. 

399 

673 

721 

509 

173 

2. 

238 

046 

103 

129 

257 

2. 

409 

933 

123 

332 

179 

2. 

252 

853 

030 

980 

263 

2. 

419 

955 

748 

490 

181 

2. 

257 

678 

574 

869 

269 

2. 

429 

752 

261 

993 

191 

2. 

281 

033 

367 

248 

271 

2. 

432 

969 

290 

877 

193 

2. 

285 

557 

309 

008  1 

277 

2. 

442 

479 

768 

999 

68 

LOGARITHMS 

N. 

Log. 

N. 

Log. 

281 

2. 

448 

706 

319 

906 

601 

2. 

778 

874 

471 

998 

283 

2. 

451 

786 

435 

523 

607 

2. 

783 

188 

691 

074 

293 

2. 

466 

867 

523 

562 

613 

2. 

787 

460 

556 

130 

307 

2. 

487 

138 

375 

477 

617 

2. 

790 

285 

164 

033 

311 

2. 

492 

760 

389 

026 

619 

2. 

791 

690 

648 

987 

313 

2. 

495 

544 

337 

650 

631 

2. 

800 

029 

359 

232 

317 

2. 

601 

069 

267 

324 

641 

2 

806 

868 

879 

634 

331 

2. 

519 

827 

993 

783 

643 

2! 

808 

210 

973 

921 

337 

2. 

627 

629 

883 

034 

647 

2. 

810 

904 

280 

666 

347 
349 

2. 

540 

329 

475 

079 

663 

2. 

814 

912 

981 

274 

2. 

642 

826 

426 

673 

659 

2. 

818 

885 

490 

409 

353 

2. 

647 

774 

138 

016 

661 

2. 

820 

201 

459 

485 

359 

2. 

655 

094 

447 

578 

673 

2- 

828 

015 

064 

225 

367 

2. 

664 

666 

064 

254 

677 

2. 

830 

588 

667 

946 

373 

2. 

571 

708 

831 

809 

683 

2. 

834 

420 

703 

630 

379 

2. 

678 

639 

209 

957 

691 

2. 

839 

477 

902 

551 

383 

2. 

583 

198 

773 

980 

701 

2. 

845 

718 

017 

237 

389 

2. 

589 

949 

601 

323 

709 

2. 

850 

646 

235 

112 

397 

2. 

598 

790 

506 

763 

719 

2. 

856 

728 

890 

383 

401 

2. 

603 

144 

372 

687 

727 

2- 

861 

634 

410 

855 

409 

2. 

611 

723 

296 

019 

733 

2. 

865 

103 

970 

639 

419 

2. 

623 

214 

m2 

971 

739 

2. 

868 

643 

643 

162 

421 

2. 

624 

282 

085 

835 

743 

9. 

870 

988 

813 

759 

431 

2. 

634 

477 

268 

999 

761 

2. 

876 

639 

937 

004 

433 

2^ 

636 

488 

016 

871 

767 

2. 

879 

095 

879 

497 

439 

2. 

642 

464 

520 

242 

761 

2. 

881 

384 

656 

769 

443 

2. 

646 

403 

726 

235 

769 

i'. 

885 

926 

339 

800 

449 

2. 

652 

246 

388 

777 

773 

2. 

888 

179 

493 

917 

467 

2. 

659. 

916 

200 

064 

787 

2. 

896 

974 

732 

358 

461 

2. 

663 

70& 

925 

389 

7a7 

2- 

901 

468 

321 

400 

■    463 

2.. 

665 

580 

994 

012 

809 

2. 

9f77 

948 

459 

773 

467 

2. 

669 

317 

88S 

008 

;  811 

2. 

909 

OQO 

864 

210 

479 

2. 

680 

335 

513 

415 

821 

2I 

914 

343 

157 

120 

487 

2. 

687 

628 

961 

120 

823 

2. 

915 

39» 

835 

203 

491 

2. 

691 

081 

487 

026 

827 

2^ 

9^17 

505 

509 

487 

499 

2. 

698 

100 

545 

623 

829 

2. 

S.18. 

654 

530 

558 

503 

2. 

701 

567 

985 

083 

839* 

2. 

923 

761 

960 

830 

1 

609 

;  2.. 

-zoe 

717 

782 

345 

853 

2. 

980 

949 

CGI 

1G3 

1 

621 

2. 

716 

837 

623 

304 

857 

2. 

932 

980 

821 

917 

523 

2. 

718 

602 

688 

873 

859 

2. 

933 

903 

163 

838 

541 

2v 

733 

197 

26B 

134 

863 

2. 

936 

010 

794 

546 

547 

2. 

737 

987 

326 

358 

877 

2. 

942 

999 

593 

360 

557 

2. 

745 

855 

195 

li92 

881 

2. 

944 

975 

908 

412 

563 

2. 

750 

508 

395 

940 

:  883 

2. 

946 

960 

703 

512 

569 

2, 

755 

112 

178 

598 

887 

2. 

947 

923 

619 

839 

571 

2. 

756 

636 

108 

333 

907 

2. 

957 

607 

287 

059 

677 

2. 

761 

175 

813 

171 

911 

2. 

959 

518 

376 

972 

587 

2. 

768 

638 

004 

465 

919 

2. 

963 

315 

513 

6C0 

593 

2. 

773 

054 

693 

364 

929 

^. 

968 

015 

713 

997 

599 

2. 

777 

427 

303 

257 

937 

2. 

971 

739 

590 

780 

1 

OF  NUMBERS. 


69 


941 

ii. 

947 

2. 

953 

2. 

967 

2. 

971 

2. 

977 

2. 

983 

2. 

991 

2. 

997 

2. 

1009 

3. 

1013 

3. 

1019 

3. 

1021 

3. 

1031 

3. 

1033 

3. 

Log. 


973  689  620  234 

976  349  979  055 

979  092  900  639 

985  426  474  084 

987  219  229  907 

989  894  559  717 

992  553  512  733 

996  073  604  003 

998  695  158  313 

003  891  170  203 

005  609  445  427 

008  174  244  007 

009  025  742  086 

013  258  660  430 

014  100  321  518 


N. 


Ix>g. 


1039 

3. 

016 

615 

647 

658 

1049 

3. 

020 

775 

488 

195 

1051 

3. 

021 

602 

716 

026 

1061 

3. 

025 

715 

383 

898 

1063 

3. 

026 

533 

264 

623 

1069 

3. 

028 

977 

705 

205 

1087 

3. 

036 

229 

513 

712 

1091 

3. 

037 

824 

749 

671 

1093 

3. 

038 

620 

157 

372 

1097 

3. 

040 

206 

627 

671 

1103 

3. 

042 

575 

612 

437 

1109 

3. 

044 

931 

546 

149 

1117 

3. 

048 

053 

173 

103 

1123 

3. 

050 

379 

756 

239 

1129 

3. 

052 

693 

942 

370 

It  is  not  necessary  to  extend  this  table,  as  the  loj^arithm  of  any 
one  of  the  higher  numbers  can  be  readily  computed  by  the  fol- 
lowing formula,  which  may  be  found  in  any  of  the  standard  works 
on  algebra,  namely  : 

Log.  (2-|-i)=log.  z-f  0.8685889638  I  -i ) 

The  result  will  be  true  to  ten  decimal  places  for  all  numbers 

over  1000,  and  true  to  twelve  decimals  for  all  numbers  over  2000. 

The  logarithms  of  composite  numbers  can   be  determined  by 

the  combination  of  logarithms  already  in  the  table,  and  the  prime 

numbers  from  the  formula. 

Thus,  the  number  3083  is  a  prime  number,  find  its  logarithm, 
true  to  ten  places  of  decimals. 

We  first  find  the  logarithm  of  3082.  By  factoring  this  num- 
ber, we  find  that  it  may  be  composed  by  the  multiplication  of  46 
into  67. 

Log.  46 1. 

Log.  67 1. 

Log.  3082 3. 

Log.  3083=3.4888321343 


662  757  8316 
826  074  3027 
488  832  1343 


Now 


0-8685889fi38 
61  6d 

We  give  a  few  additional  prime  numbers  : 


1151 
1153 
1163 
1171 
1181 
1187 
1193 
1201 
1213 
1217 


1223 
1229 
1231 
1237 
1249 
1259 
1277 
1279 
1283 
1289 


1291 
1297 
1301 
1303 
1307 
1319 
1321 
1327 
1361 
1367 


1373 
1381 
1399 
1409 
1423 
1427 
1429 
1433 
1439 
1447 


1461 
1453 
1459 
1471 
1481 
1483 
1487 
1489 
1493 
1499 


1511 
1523 
1531 
1543 
1549 
1553 
1559 
1667 
1671 
1579 


70 

AUXILIARY    LOGARITHMS. 

AUXILIARY    LOGARITHMS*. 

N. 

Log. 

N. 

Log. 

1.  009 

0.  003  891  170  2031 

1.  0009 

0.  000  390  576  304] 

1.  008 

0.  003  461  627  188 

1.  0008 

0.  000  347  233  698 

1.  007 

0.  003  030  465  635 

1.  0007 

0.  000  303  836  798 

1.  006 

0.  002  597  985  739 

1.  0006 

0.  000  260  435  661 

1.  005 

0.  002  166  071  750 

[►A 

1.  0005 

0.  000  217  099  966 

Ib 

1.  004 

0.  001  733  722  804 

1.  0004 

0.  000  173  690  053 

1.  003 

0.  001  300  943  017 

1.  0003 

0.  000  130  268  803 

1.  002 

0.  000  867  721  529 

1.  0002 

0.  000  086  850  213 

1.  001 

0.  000  434  077  479  J 

1.  0001 

0.  000  043  427  277^ 

— 

N. 

Log. 

1.     00009 

0. 

000  039  084  741         1 

1.     00008 

0. 

000  034  742  166 

1.     00007 

0. 

000  030  399  546 

1 .     00006 

0. 

000  026  066  884 

1.     00005 

0. 

000  021  714  178 

1.     00004 

0. 

000  017  371  430 

1.     00003 

0. 

000  013  028  638 

C 

1.     00002 

0. 

000  008  685  802 

1.     00001 

0. 

000  004  342  923  (a) 

1.     000001 

0. 

000  000  434  294  (b) 

1.     0000001 

0. 

000  000  043  429  (c) 

1.     00000001 

0. 

000  000  004  343  (d) 

1.     000000001 

0. 

000  000  000  434  (e) 

1.     0000000001 

0. 

000  000  000  043  (f)J 

N» 

imber. 

Log. 

0.  43. 

12944819 

—1.     637  784  298 

This  decimal  number  is  the  modulus  of  our  system  of  logarithms.  | 

Its  loga 

rithm  is  very  useful  in  correcting  other  logarithms,  as  may 

be  seen 

in  the  Chapter  on  Logarithms. 

TAULfci  V.                                                          TAULE  VII. 

"! 

Dip  of  the  Sea  Horizon.                         Mean  Refraction  of  Celestial  Objects. 

'.^S 

b2 

\^'» 

g2 

Alt 

Rcfr.  ||  Alt.|Uefr 

,    Alt 

llffr.;|  Alt. 

Refr. 

Alt. 

Refr 

»d5- 

o-o 

?.| 

§0 

0 

/        rf\  0 

f  f    1 

"o 

/     II 

1  '^ 

/  1    II 

0 

—FT 

0^ 

5'S, 

55^ 

5- a 

0  c 

33     OlllO  C 

5  16 

>( 

2  35 

32  C 

)1  30 

67 

24 

P^ 

H 

^2. 

P  D* 

IC 

31  32 

IC 

6  10 

'       1( 

2  24 

40  1  29 

68 

23 

~t     If 

„ 

2C 

29  60 

20 

5  05 

2( 

2  22 

33  C 

1  28 

69 

22 

1 

0  59 

38 

6    4 
6  18 
6  32 
6  45 

6  58 

7  10 
7  12 

3C 

28  23 

30 

5  00 

3( 

2  21 

2C 

1  26 

70 

21 

2 

I  24 
1  42 

41 

44 

4C 

27  OG 

40 

4  66 

4( 

2  29 

4C 

1  26 

71 

19 

3 

5C 

25  42 

60 

4  61 

5( 

2  28 

34  C 

1  24 

72 

18 

4 

1  58 

2  12 
2  25 

47 
50 
53 

1  C 

24  29 

11  0 

4  47 

21  ( 

2  27 

2C 

1  23 

73 

17 

6 

10 

23  20 

10 

4  43 

IC 

2  26 

4C 

1  22 

74 

16 

6 

2C 

22  15 

20 

4  39 

2C 

2  25 

35  C 

1  21 

76 

15 

7 

2  36 

66 

3C 

21  15 

30 

4  34 

3C 

2  24 

20 

1  20 

76 

14 

8 

2  47 

69 

7  24 

4C 

20  18 

40 

4  31 

4C 

2  23 

40 

1  19 

77 

13 

9 

2  57 

62 

'     745 

50 

19  25 

60 

4  27 

5C 

2  21 

36  0 

1  18 

78 

12 

10 

3  07 

66 

7  66 

2  0 

18  35 

12  0 

4  23 

2^  C 

2  20 

30 

1  17 

79 

11 

11 

3  16 

68 

8  07 

10 

17  48 

10 

4  20 

IC 

2  19 

37  0 

1  16 

80 

10 

12 

3  25 

71 

8  18 

20 

17  04 

20 

4  16 

20 

2  18 

30 

1  14 

81 

9 

13 

3  33 

74 

8  28 

30 

16  24 

30 

4  13 

30 

2  17 

38  0 

1  13 

82 

8 

14 

3  41 

77 

8  38 

40 

16  45 

40 

409 

40 

2  16 

30 

1  11 

83 

7 

15 

3  49 

80 

8  48 

50 

15  09 

60 

4  06 

50 

2  15 

39  0 

1  10 

34 

6 

16 

3  56 

83 

8  58 

3  0 

14  34 

13  0 

4  03 

23  0 

2  14 

30 

1  09 

85 

5 

17 

4  04 

86 

9  08 

10 

14  04 

10 

4  00 

10 

2  13 

400 

1  08 

86 

4 

18 

4  11 

89 

9  17 

20 

13  34 

20 

3  57 

20 

2  12 

30 

1  07 

87 

3 

19 

4  17 

92 

9  26 

30 

13  06 

30 

3  54 

30 

2  11 

41  0 

1  05 

88 

2. 

20 

4  24 

95 

9  36 

40 

12  40 

49 

3  61 

40  2  10 

30 

1  04 

89 

1 

21 

4  31 

98 

9  45 

60 

12  15 

60 

3  48 

50  2  09 

42  0 

1  03 

90 

0 

22 

4  37 

JXl 

9  54 

4  0 

11  61 

14  0 

3  45 

24  02  08 

30 

1  02 

23 

4  43 

104 

10  02 

10 

11  29 

10 

3  43 

102  07 

43  0 

1  01 

24 

4  49 

107 

10  11 

20 

11  08 

20 

3  40 

202  06 

30 

1  00 

26 

4  55 

110 

10  19 

30 

10  48 

30  3  38 

30 

2  06 

44  0 

0  69 

26 

5  01 

III 

10  28 

40 

10  29 

40 

3  35 

40 

2  04 

80 

0  58 

27 

5  07 

116 

10  36 

50 

10  11 

60 

3  33 

60 

2  03 

45  0 

0  57 

28 
29 

5  13 
5  18' 

119 

122 

10  44 
10  62 

5  0 

9  64 

15  0 

3  30 

25  0 

2  02 

30 

0  66 

30 

5  241 

125 

11  00 

10 

9  38 

10 

3  28 

10 

2  01 

46  0 

0  55 

31 

5  29 1 

128 

11  08 

20 

9  23 

20 

3  26 

20 

2  00 

30 

0  54 

82 

5  34 

131 

11  16 

30 

9   08: 

30 

3  24 

30 

1  59 

47  0 

0  53 

33 

5  39 

134 

11  24 

40 

8  54 

40 

3  21 

40 

1  58 

30 

0  62 

34 

5  44 

137 

11  31 

50 

8  41 

50 

3  19 

50 

1  67 

48  0 

0  61 

35 

5  49I 

140 

11    RQ 

6  0 

8  28 

16  0 

3  17 

26  0 

1  66 

30 

0  60 

10 

8  16 
8  03 

10 
20 

3  16 
3  12 

10 
20 

1  65 
1  56 

49  0 
30 

0  49 
0  49 

20 

TABLE  VI. 

30 

7  15 

30 

3  10 

30 

154 

60  0  0  48 

Dip  of  the  Sea  Horizon  at 

40 

7  40! 

40 

l^ 

40 

1  63 

30  0  47 

different  Distances  from  it. 

50 
7  0 

7  30: 
7  2O1 

60 
17  0 

3  06 
3  04 

50 
27  0 

1  62 
1  61 

51  0 
30 

0  46 
0  45 

10 

fT     1   li 

10 
20 
30 

3  03 
3  01 

15 
30 

1  50 
1  49 

52  0 
30 

53  0 

0  44 
)  44 

Dist. 

Hight  of  Eye  in  i't.l 

IV     1     xii 

20   T  noJ 

in 

Miles. 

5 

10 

16   2 

0J25 

30 

30 
40 

6  63| 

2  69 

46 
28  0 

1  48 

0  43 

T 

~7~ 

~T   ~ 

-  — 

~ 

6  45| 

40 

2  67 

1  47 

30 

O  42 

i 

11 

22 

34  4 

5 '56 

68 

50 

6  37i 

60 

2  56 

15 

1  46  54  0 

1  45  56  0 

3  41 

6 

17  2 

2 '28 

34 

8  0 

6  29i 

18  0 

2  64l 

30 

}  40 

4 

8 

12  1 

5J19 

23 

10 

6  22| 

10 

2  52 

45 

1  44  56  0 

}38 

I 

4 

6 

9  li 

2!l5 

17 

20 

6  15 

20 

2  51i 

29  0 

1  42  57  0 

)  37 

U 

3 

5 

7    < 

)  12 

14 

30 

6  08 

30 

2  49 

20 

1  4ll58  0 

3  35 

H 

3 

4 

6    I 

I    9 

12 

40 

6  01 

40 

2  47 

40 

I  40|!59  0 

)  34 

2 

2 

3 

5    ( 

)    8 

10 

50 

5  56 

50' 

2  46 

30  0 

I  38|;60  0 

)  33 

2i 

2 

3 

6    ( 

5    7 

8 

9  0 

5  98 

19  0; 

2  44 

20 

I  37JI61  0 

)  32 

3 

2 

3 

4    I 

)    6 

7 

10 

5  42 

10  i 

243; 

40 

I  361^2  0 

)  80 

3i 

2 

3 

4    £ 

)    6 

6 

20 

6  46 

20^ 

2  41 

31  0 

I  35;  63  0 
I  33' 64  0 
I  32  165  0( 

)  29 

4 

2 

3 

4    ^ 

I    5 

6 

30 

6  41 

30  5 

J  40 

20 

)28 

5 

2 

3 

4    4 

I    5 

5 

40 

6  26 

40  S 

\  38 

40 

)  26 

6 

2 

3 

4    4 

I    5    6| 

60 

6  20! 

60  2  37il32  0| 

I  3ll^6  0( 

)  25 

1 

1 

>w. 


1 


3f43 


37 


-.^ 


^'»  ^^ 


